QA 


, 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFOF 

O-TKT  OF" 


Received 
Accession  No.£> 


.    Class  No. 


EJajj  cinfr  @ljoms0n's   Series. 

RUDIMENTS 

UF 

ARITHMETIC; 


CONTAINING 


NUMEROUS    EXERCISES 


FOR  THE 


ti  L  A  T  E   AN  D,,RL  A*,C  KB  O  A  E  D 

FOK  BEGINNERS. 
BY  JAMES  B.  THOMSON,  LL.D., 

^UTPOR    OF    MENTAL    ARITHMETIC;    EXJ?UCISKS    TN    ARITHMETICAL   ANALV8I8  4 

TACTICAL  ARITHMKTIG;  HIOIIKR  ARITHMKTIC;  EDITOR  OF  DAY'S 
OOL  ALOKBRA;  LBOENDRK'S  OEO^KTKY,  ETC. 


REVISED 


NEW  TOEK: 
CLARK  &  MAYNARD,  5  BARCLAY  ST. 

CHICAGO  :   S.  C.  GRIGGS  &   CO. 

1808. 


Entered  according  to  Act  of  Congress,  in  the  year  1^53    by 

JAMES  B.  THOMSON, 
In  the  Clerk's  Office  far  the  Southern  District  of  New  York 


PREFACE. 


EDUCATION,  in  its  comprehensive  sense,  is  the  business  of 
life.  The  exercises  of  the  school-room  lay  the  foundation  ; 
the  superstructure  is  the  work  of  after  years.  If  these  exer- 
cises are  rightly  conducted,  the  pupil  obtains  the  rudiments 
of  science,  and  what  is  more  important,  he  learns  how  to 
study,  how  to  think  and  reason,  and  is  thus  enabled  to  appro- 
priate the  means  of  knowledge  to  his  future  advancement. 
Any  system  of  instruction,  therefore,  which  does  not  embrace 
these  objects,  which  treats  a  child  as  a  mere  passive  recipient, 
is  palpably  defective.  It  is  destitute  of  some  of  the  most 
essential  means  of  mental  development,  and  is  calculated  to 
produce  pigmies,  instead  of  giant  intellects. 

The  question  is  often  asked,  "  What  is  the  best  method  of 
proceeding  with  pupils  commencing  the  study  of  Arithmetic, 
or  entering  upon  a  new  rule  ?" 

The  old  method. — Some  teachers  allow  every  pupil  to  cipher 
"  on  his  own  hook;"  to  go  as  fast,  or  as  slow  as  he  pleases, 
without  reciting  a  single  example  or  rule,  or  stopping  to  in- 
quire the  "  why  and  the  wherefore"  of  a  single  operation. 
This  mode  of  teaching  is  a  relic  of  by-gone  days,  and  is  prima 
facie  evidence,  that  those  who  practice  it,  are  "behind  the 
spirit  of  the  times. 

Another  method. — Others  who  admit  the  necessity  of  teach- 
ing arithmetic  in  classes,  send  their  pupils  to  their  seats,  and 
tell  them  to  "study  the  rule."  After  idling  away  an  hour 
or  more,  up  goes  one  little  hand  after  another  with  the  de- 
spairing question  : — u  Please  to  show  me  how  to  do  this  sum, 
sir  ?"  The  teacher  replies,  "Study  the  rule; — that  will  tell 
you."  At  length,  to  silence  their  increasing  importunity,  he 
takes  the  slate,  solves  the  question,  and,  without  a  word  of 


IV  PREFACE. 

explanation,  returns  it  to  its  owner.  He  thus  goes  through 
the  class.  "When  the  hour  of  recitation  comes,  the  class  is 
not  prepared  with  the  lesson.  They  are  sent  to  their  seats 
to  make  another  trial,  which  results  in  no  better  success. 
And  what  is  the  consequence  ?  They  are  discouraged  and 
disgusted  with  the  study. 

A  more  excellent  way. — Other  teachers  pursue  a  more  ex- 
cellent way,  especially  for  young  pupils.  It  is  this : — The 
teacher  reads  over  with  the  class  the  preliminary  explanations, 
and  after  satisfying  himself  that  they  understand  the  mean- 
ing of  the  terms,  he  calls  upon  one  to  read  and  analyze  the 
first  example,  and  set  it  down  upon  the  blackboard,  while 
the  rest  write  it  upon  their  slates.  The  one  at  the  board 
then  performs  the  operation  audibly,  and  those  with  their 
slates  follow  step  by  step. 

Another  is  now  called  to  the  board  and  requested  to  set 
down  the  second  example,  while  the  rest  write  the  same 
upon  their  slates,  and  solve  it  in  a  similar  manner.  He  then 
directs  them  to  take  the  third  example,  and  lets  them  try 
their  own  skill,  giving  each  such  aid  as  he  may  require.  In 
this  way  they  soon  get  hold  of  the  principle,  and  if  now  sent 
to  their  seats,  will  master  the  lesson  with  positive  delight. 

As  to  assistance,  no  specific  directions  can  be  given  which 
will  meet  every  case.  The  best  rule  is,  to  afford  the 
learner  just  that  kind  and  amount,  which  will  secure  the 
greatest  degree  of  exertion  on  his  part.  Less  than  this  dis- 
courages; more,  enervates. 

In  conclusion,  we  would  add,  that  this  elementary  work 
was  undertaken  at  the  particular  request  of  several  eminent 
practical  teachers,  and  is  designed  to  fill  a  niche  in  primary 
schools.  It  presents,  in  a  cheap  form,  a  series  of  progressive 
exercises  in  the  simple  ftnd  compound  rules,  which  are 
adapted  to  the  capacities  of  beginners,  and  are  calculated  to 
form  habits  of  study,  awaken  the  attention,  and  strengthen 
the  intellect. 

J.  B.  THOMSON. 
YOEK,  January,  1853. 


CONTENTS. 


SECTION   I. 

Pago 

ARITHMETIC  defined, •? 

Notation, 7 

Roman  Notation, 7 

Arabic  Notation,          -        -        - 9 

Numeration, 12 

SECTION   II 
ADDITION  defined,         ...  ...-16 

When  the  sum  of  a  column  does  not  exceed  9,         •         -         -  18 
When  the  sum  of  a  column  exceeds  9,  -         -         -        -        -       19 

General  Rule  for  Addition,      • 20 

SECTION   III. 

SUBTRACTION  defined, 27 

When  a  figure  in  the  lower  No.  is  smaller  than  that  above  it,  -  28 
When  a  figure  in  the  lower  No.  is  larger  than  that  above  it,  29 

Borrowing  10, SO 

General  Rule  for  Subtraction,      -         -        -        -        -        -81 

SECTION   IV. 

MULTIPLICATION  defined, 36 

When  the  multiplier  contains  but  one  figure,        -         -  39 

When  the  multiplier  contains  more  than  one  figure,  -  -  41 
General  Rule  for  Multiplication,  ------  43 

To  multiply  by  10,  100,  1000,  <fec., 45 

When  there  are  ciphers  on  the  right  of  the  multiplier,  -  46 
When  there  are  ciphers  on  the  right  of  the  multiplicand,  -  47 
When  there  are  ciphers  on  the  right  of  both,  -  48 

SECTION   V. 

DIVISION  defined,  -  -  -  -  .  -  -  --49 
Short  Division, 52 


VI  x  CONTENTS. 

Page 
Rule  for  Short  Division, 54 

Long  Division,  -  ...  -56 

Difference  between  Short  and  Long  Division,        -  -     57 

Rule  for  Long  Division, -58 

To  divide  by  10,  100,  1000,  Ac.,     -  -     61 

When  there  are  ciphers  on  the  right  of  the  divisor,  -         -         62 

Arithmetical  Terms, 63 

greatest  Common  Divisor, 64 

Least  Common  Multiple,        • 65 

SECTION  VI. 

FRACTIONS, 67 

Reduction  of  Fractions, 69 

Contraction  by  Cancellation, 72 

Addition  of  Fractions,  -        -         -        -        •         •        -        -76 

Subtraction  of  Fractions,  -         - 78 

Multiplication  of  Fractions, 80 

Division  of  Fractions, 84 

SECTION   VII. 

TABLES  in  Compound  Numbers, 88 

Paper  and  Books,      ........99 

Tables  of  Aliquot  Parts, 101 

SECTION    VIII. 

ADDITION  of  Federal  Money,     • 102 

Subtraction  of  Federal  Money,      -        -        -        -        -         -106 

Multiplication  of  Federal  Money,       -        -        -        -        -    '  107 

Division  of  Federal  Money, 108 

SECTION   IX. 
REDUCTION, 110 

Rule  for  Reduction  Descending, Ill 

Rule  for  Reduction  Ascending, 114 

Compound  Addition,      -        -        -        -        -        -        -        -120 

Compound  Subtraction, 122 

Compound  Multiplication, 124 

Compound  Division, 125 

Miscellaneous  Exercises,        -        -        •        -        •        •         -127 
Answers  to  Examples,       -        -        -        *         «        -        -133 


ARITHMETIC. 


SECTION    I. 

"~  ART.  !•  ARITHMETIC  is  the  science  of  numbers. 

Any  single  thing,  as  a  peach,  a  rose,  a  book,  is  called  a 
unit,  or  one ;  if  another  single  thing  is  put  with  it,  the 
collection  is  called  two  ;  if  another  still,  it  is  called  three  ; 
if  an  other,  four  ;  if  another,  five,  &c. 

The  terms,  one,  two,  three,  four,  <&c.,  are  the  names  of 
numbers.  Hence, 

2.  NUMBER  signifies  a  unit,  or  a  collection  of^units.  . 
— ^  Numbers   are  expressed   by 'words,  by'  letters,  and  by 

figures.  I 

3.  NOTATION  is  £Ae  ar2  of  expressing  numbers  by  letters 
or  figures.     There  are  two  methods  of  notation  in  use,  the 
Roman  and  the  Arabic^ 

I.   ROMAN  NOTATION. 

4»  The  Roman  Notation  is  the  method  of  expressing 
-  numbers  by  letters  ;  and  is  so  called  because  it  was  invented 
by  the  ancient  Romans.     It  employs  seven  capital  letters, 
via  :  I,  V,  X,  L,  C,  D,  M. 

When  standing  alone,  the  letter  I,  denotes  one  ;  V,  five  ; 
X,  ten  ;  L,  fifty  ;  C,  one  hundred  ;  D,  five  hundred  ;  M, 
one  thousand. 


QUEST.— 1.  What  is  Arithmetic?  What  is  a  single  thing  called  ?  If  an- 
other is  put  with  it,  what  is  the  collection  called  ?  If  another,  what  ?  What 
are  the  terms  one,  two,  three,  &c.  ?  2.  What  then  is  number?  How  aro, 
numbers  expressed?  '.1.  What  is  Notation  ?  How  many  methods  of  notation 
are  in  use?  4.  What  is  the  Roman  notation?  Why  so  called?  How  nnny 
letters  does  item  piny?  What  does  the  letter  I,  denote?  V?  X?T.?  CV  !>r  >.».-» 


8  NOTATION.  [SECT.  I. 

5.  To  express  the  intervening  numbers  from  to  one  a 
thousand,  or  any  number  larger  than  a  thousand,  we  re- 
sort to  repetitions  and  various  combinations  of  these  let- 
ters, as  may  be  seen  from  the  following 

TABLE. 


I         denotes 

one. 

XXXI  denotes  thirty-one. 

II              " 

two. 

XL 

"     forty. 

III 

three. 

XLI 

"     forty-one. 

IV 

four. 

L 

"     fifty. 

V               « 

five. 

LI 

"     fifty-one. 

VI 

six. 

LX 

"     sixty. 

VII 

seven. 

LXI   / 

"    sixty-one. 

VIII 

eight. 

LXX 

"     seventy. 

IX 

nine. 

LXXX  \» 

"     eighly. 

X 

ten. 

XC 

"     ninety. 

XI 

eleven. 

XCI 

"     ninety-one. 

XII 

twelve. 

C 

"     one  hundred. 

XIII 

thirteen. 

CI     / 

''     one  hund.  and  one. 

XIV          « 

fourteen. 

CIV 

l<     one  hund.  and  four. 

XV 

fifteen. 

ex 

"     one  hund.  and  ten.' 

XVI 

sixteen. 

CO 

"     two  hundred. 

XVII        « 

seventeen. 

ccv 

"     two  hund.  and.five. 

XVIII      " 

eighteen. 

ccc 

"     three  hundred. 

XIX         " 

nineteen. 

cccc 

"     four  hundred. 

XX  v  .      " 

twenty. 

D 

"     five  hundred. 

XXI    ,    " 

twenty-one. 

DC 

"     six  hundred. 

XXII  1    « 

twenty-two. 

DCC 

"     seven  hundred. 

XXIII      « 

twenty-three. 

DCCC 

"     eight  hundred. 

XXIV      « 

twenty-  four. 

DCCCC 

"     nine  hundred. 

XXV 

twenty-five. 

M 

"     one  thousand. 

XXVI       " 

twenty-six. 

MC 

"     one   thousand  and 

XXVII     « 

twenty-seven. 

one  hundred. 

XXVIII  « 

twenty-eight. 

MM 

"     two  thousand. 

XXIX      " 

twenty-nine. 

MDCCCL 

"     one  thousand  eight 

XXX        " 

thirty. 

hundred  and  fifty- 

QUBST.— 5.  What  do  the  letters  IV,  denote?  VI?  VIII?  IX?  XI?  XIV? 
XVI?  XVIH?  XIX?  XXIV?  XL?  LXXX?  XC?  CIV?  Express 
seven  by  letters  on  the  slate  or  black-board.  How  express  eleven  ?  Thirteen  ? 
Twtjnty-five ?  Nineteen?  Forty-four?  Eighty-seven?  Ninety-nine? 


ARTS.  5 — 7.]  NOTATION.  9 

OBS.  1.  Every  time  a  letter  is  repeated,  its  value  is  rep<  ,ted. 
Thus,  the  letter  I,  standing  alone,  denotes  one  ;  II,  two  <>nes  or  two, 
<fcc.  So  X  denotes  ten  ;  XX,  twenty,  <fec. 

2.  When  two  letters  of  different  value  are  joined  togeth'er,  if  the 
less  is  placed  before  the  greater,  the  value  of  the  greater  is  dimin- 
ished as  many  units  as  the  less  denotes ;  if  placed  after  the  greater, 
the  value  of  the  greater  is  increased  as  many  units  as  the  less  de- 
notes. Thus,  V  denotes  five ;  but  IV  denotes  only  four ;  and  VI, 
six.  So  X  denotes  ten ;  IX,  nine  ;  XI,  eleven. 

Note. — The  questions  on  the  observations  may  be  omitted,  by 
beginners,  till  review,  if  deemed  advisable  by  the  teacher. 

II.   AKABIC  NOTATION. 

6.  The  Arabic  Notation  is  the  method  of  expressing 
"numbers  by  figures  ;  and  is  so  called  because  it  is  supposed 
to  have  been  invented  by  the  Arabs.    -It  employs  the  fol- 
lowing ten  characters  or  figures,  viz  : 

123456789/0 
one,      two,      three,    four,      five,       six,      seven,    eight,    nine,    naught. 

OBS.  1.  The  first  nine  are  called  significant  figures,  because  each 
one  always  expresses  a  value,  or  denotes  some  number.  They  are 
also  called  digits,  from  the  Latin  word  digitus,  signify  ing  a  finger; 
because  the  ancients  used  to  count  on  their  fingers. 

2.  The  last  one  is  called  naught,  because  when  standing  alone, 
it  expresses  nothing,  or  the  absence  of  number.  It  is  also  called 
cipher  or  zero. 

7  •  All  numbers  larger  than  9,  are  expressed  by  different 
combinations  of  these  ten  figures.  For  example,  to  express 
ten,  we  use  the  1  and  0,  thus  10  ;  to  express  eleven,  we 
use  two  Is,  thus  11  ;  to  express  twelve,  we  use  the  1  and 
2,  thus  12,  &c. 

QUEST.—  Obs.  What  is  the  effect  of  repeating  a  letter?  If  a  letter  of  less 
value  is  placed  before  another  of  greater  value,  what  is  the  effect?  If  placed 
after,  what  ?  6.  What  is  the  Arabic  notation  ?  Why  so  called  ?  How  many 
figures  does  it  employ?  What  are  their  names?  Ob*.  What  are  the  first  nine 
called?  Why?  What  else  are  they  sometimes  called?  What  is  the  last  one 
called?  Why?  7.  How  are  numbers  larger  than  nine  expressed  ?  Express 
ten  by  figures.  Eleven.  Twelve.  Fifteen 


10 


NOTATION. 


[SECT.  J. 


The  method  of  expressing  numbers  by  figures  from 
one  to  a  thousand,  may  be  seen  from  the  following 


TABLE. 


1,  one. 

36,  thirty-six. 

71,  seventy-one. 

2,  two. 

37,  thirty-seven. 

72,  seventy-two. 

3,  three. 

38,  thirty-eight. 

73,  seventy-three, 

4,  four. 

39,  thirty-nine. 

74,  seventy-four. 

5,  five. 

40,  forty. 

75,  seventy-five. 

6,  six. 

41,  forty-one. 

76,  seventy-six. 

7,  seven. 

42,  forty-two. 

77,  seventy-seven. 

8,  eight. 

43,  forty-three. 

78,  seventy-eight. 

9,  nine. 

44;  forty-  four. 

79,  seventy-nine. 

10,  ten. 

45,  forty-five. 

80,  eighty. 

11,  eleven. 

46,  forty-six. 

81,  eighty-one. 

12,  twelve. 

47,  forty-seven. 

82,  eighty-two. 

13,  thirteen. 

.48,  forty-eight. 

83,  eighty-three. 

14,  fourteen. 

49,  forty-nine. 

84,  eighty-  four.    ' 

15,  fifteen. 

50,  fifty. 

85,  eighty-five. 

16,  sixteen. 

51,  fifty-one. 

86,  eighty-six. 

17,  seventeen. 

52,  fifty-two. 

87,  eighty-seven. 

18,  eighteen. 

53,  fifty-three. 

88,  eighty-eight. 

19,  nineteen. 

54,  fifty-  four. 

89,  eighty-nine. 

20,  twenty. 

55,  fifty-five. 

90,  ninety. 

2t,  twenty-one. 

56,  fifty-six. 

91,  ninety-one. 

22,  twenty-two. 

57,  fifty-seven. 

92,  ninety-two. 

23,  twenty-three. 

58,1fifty-eight. 

93,  ninety-three. 

24,  twenty-  four. 

59,  fifty-nine. 

94,  ninety-  four. 

25,  twenty-five. 

60,  sixty. 

95,  ninety-five. 

26,  twenty-six. 

61,  vixty-one. 

96,  ninety-six. 

27,  twenty-seven. 

62,  sixty-two. 

97,  ninety-seven. 

28,  twenty-eight. 

63,  sixty-three. 

98,  ninety-eight. 

29,  twenty-nine. 

64,  sixty-  four. 

99,  ninety-nine. 

30,  thirty. 

65,  sixty-five. 

100,  one  hundred. 

31,  thirty-one. 

66,  sixty-six. 

200,  two  hundred. 

32,  thirty-two. 

67,  sixty-seven. 

300,  three  hundred. 

33,  thirty-three. 

68,  sixty-eight. 

400,  four  hundred. 

34,  thirty-  four. 

69.  sixty-nine. 

900,  nine  hundred. 

35.  thirty-five. 

70,  seventy. 

1000,  one  thousand. 

QUEST.— How  express  fifteen?  Twenty-five?  Forty-seven?  Thirty-six? 
Seventy-three?  One  hundred  and  one?  One  hundred  and  ten?  One  hundred 
and  twenty  ?  Two  hundred  and  fifteen  ? 


ARTS.  8 — ll.j  NOTATION.  11 

8»  It  will  be  perceived  from  the  foregoing  table,  that 
the  same  figures,  standing  in  different  places,  have  differ- 
ent values. 

When  they  stand  alone  or  in  the  right  hand  place,  they 
express  units  or  ones,  which  are  called  units  of  the  first 
order, 

When  they  stand  in  the  second  place,  they  express  tens, 
which  are  called  units  of  the  second  order. 

When  they  stand  in  the  third  place,  they  express  hun- 
dreds, which  are  called  units  of  the  third  order. 

When  they  stand  in  the  fourth  place,  they  express 
thousands,  which  are  called  units  of  the  fourth  order,  <fec. 

For  example,  the  figures  2,  3,  4,  and  5,  when  arranged 
thus,  2345,  denote  2  thousands,  3  hundreds,  4  tens,  and  5 
units  ;  when  arranged  thus,  5432,  they  denote  5  thousands, 
4  hundreds,  3  tens,  and  2  units. 

9»  Ten  units  make  one  ten,  ten  tens  make  one  hundred, 
and  ten  hundreds  make  one  thousand,  <fec. ;  that  is,  ten  of 
any  lower  order,  are  equal  to  one  in  the  next  higher  order. 
Hence,  universally, 

1C.  Numbers  increase  from  right  to  left  in  a  tenfold 
ratio  ;  that  is,  each  removal  of  a  figure  one  place  towards 
the  left,  increases  its  value  ten  times. 

1  !•  The  different  values  which  the  same  figures  have, 
are  called  simple  and  local  values. 

The  simple  value  of  a  figure  is  the  value  which  it  ex- 
presses when  it  stands  alone,  or  in  the  right  hand  place. 

QUEST.— 8.  Do  the  same  figures  always  have  the  same  value  ?  When  stand- 
ing alone  or  in  the  right  hand  place,  what  do  they  express?  What  do  they 
express  when  standing  in  the  second  place?  In  the  third  place?  In  tho 
fourth?  9.  How  many  units  make  one  ten?  How  many  tens  make  a  hun- 
dred ?  How  many  hundreds  make  a  thousand  ?  Generally,  how  many  of  any 
ower  order  are  required  to  make  one  of  the  next  higher  order  ?  10.  What  is 
the  general  law  by  which  numbers  increase?  What  is  the  effect  upon  the  v:tlue 
of  a  figure  to  remove  it  one  place  towards  the  left  ?  11.  What  are  the  differ- 
ent values  of  the  same  figure  called  ?  What  is  the  simple  value  of  a  fkptro  1 
VVhnt  the  local  value  ? 


12 


NUMERATION. 


[SECT.  I 


The  simple  value  of  a  figure,  therefore,  is  the  number 
which  its  name  denotes. 

The  local  value  of  a  figure  is  the  increased  value  which 
it  expresses  by  having  other  figures  placed  on  its  right. 
Hence,  the  local  value  of  a  figure  depends  on  its  locality, 
or  the  place  which  it  occupies  in  relation  to  other  num- 
bers with  which  it  is  connected.  (Art.  8.) 

OBS.  This  system  of  notation  is  also  called  the  decimal  system^ 
because  numbers  increase  in  a  tcnfolg  ratio.  The  term  decimal  is 
derived  from  the  Latin  word  dccem,  which  signifies  ten. 

NUMERATION. 

12.  The  art  of  reading  numbers  when  expressed  by 
figures,  is  called  Numeration. 

NUMERATION    TABL^. 


I 

3   <» 

S  i 


e  "s 

"2       CB 
5       G 


c  -^ 

«   M 


123       8 


fi 

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Period  V.      Period  IV.      Period  III.       Period  II.        Period  I. 
Trillions.       Billions.          Millions.       Thousands.         Units. 

13.  The  different  orders  of  numbers  are  divided  into 
periods  of  three  figures  each,  beginning  at  the  right  hand. 

QUEST. — Upon  what  does  the  local  value  of  a  figure  depend  (i  Obs.  What  ia 
this  system  of  notation  sometimes  called  'i  Why  't  12.  What  is  Numeration  'i 
Repeat  the  numeration  table,  beginning  at  the  right  hand.  What  is  the  first 
place  on  the  right  called ?  The  second  place?  The  third?  Fourth?  Fifth? 
Sixth?  Seventh?  Eighth?  Ninth?  Tenth,  &c.?  13.  How  are  the  orders  of 
numbers  divided  ? 


ARTS.  12 — 14.]  NUMERATION.  13 

The  first,  or  right  band  period  is  occupied  by  units,  tens, 
hundreds,  and  is  called  units'  period ;  the  second  is  oc- 
cupied by  thousands,  tens  of  thousands,  hundreds  of 
thousands,  and  is  called  thousands'  period,  &c. 

The  figures  in  the  table  are  read  thus :  One  hundred 
and  twenty-three  trillions,  eight  hundred  and  sixty-one 
billions,  five  hundred  and  eighteen  millions,  nine  hundred 
and  twenty-four  thousand,  two  hundred  and  sixty-three. 

1 4«  To  read  numbers  which  are  expressed  by  figures. 

Point  them  off  into  periods  of  three  figures  each  ;  then, 
beginning  at  the  left  hand,  read  the  figures  of  each  period 
as  though  it  stood  alone,  and  to  the  last  figure  of  each,  add 
the  name  of  the  period. 

OBS.  1.  The  learner  must  be  careful,  in  pointing  off  figures,  always 
to  begin  at  the  rig/it  hand ;  and  in  reading  them,  to  begin  at  the 
Left  hand. 

2.  Since  the  figures  in  the  first  or  right  hand  period  always  de- 
note units,  the  name  of  the  period  is  not  pronounced.  Hence,  in 
reading  figures,  when  no  period  is  mentioned,  it  is  always  under- 
stood to  be  the  right  hand,  or  units'  period. 

EXERCISES    IN    NUMERATION. 

Note. — At  first  the  pupil  sbould  be  required  to  apply  to  each  fig- 
ure the  name  of  the  place  which  it  occupies.  Thus,  beginning  at 
the  right  hand,  he  should  say,  "  Units,  tens,  hundreds,"  &c.,  and 
point  at  the  same  time  to  the  figure  standing  in  the  place  which  he 
mentions.  It  will  be  a  profitable  exercise  for  young  scholars  to 
write  the  examples  upon  their  slates  or  paper,  then  point  them  off 
into  periods,  and  read  them. 


QUEST.— Wh&t  is  the  first  period  called  ?  By  what  is  it  occupied  ?  What  is 
the  second  period  called?  By  what  occupied?  What  is  the  ttiird  period 
called  V  By  what  occupied  ?  What  is  the  fourth  called  ?  By  what  occupied  ? 
What  is  the  fifth  called?  By  what  occupied?  14.  How  do  you  read  numbers 
expressed  by  figures?  Obs.  Where  begin  to  point  them  off?  Where  to  read 
them?  Do  you  pronounce  the  name  of  the  right  hand  period?  When  uo 
period  is  named,  wlml  is  understood  ? 


14 


NUMERATION. 


Read  the  following  numbers : 


[SECT.  I. 


Ex.  1. 

97 

16. 

12642 

31.      7620 

2. 

110 

17. 

20871 

32.      8  040 

3. 

256 

18. 

17046 

33.      9638 

4. 

307 

19. 

43201 

34.     11000 

5. 

510 

20. 

80600 

35.     12100 

6. 

465 

21. 

4203 

36.     14020 

7. 

1248 

22. 

65026 

37.     10001 

8. 

2381 

23. 

78007 

38.      5  020 

9. 

4026 

24. 

90210 

39.     18022 

10. 

6420 

25. 

5025 

40.     30401 

11. 

8600 

26. 

69008 

41.      2506 

12. 

7040 

27. 

100000 

42.    402321 

13. 

8000 

28. 

125236 

43.    '65007 

14. 

9007 

29. 

6005 

44.    750026 

15. 

10000 

30. 

462400 

45.    804420 

46.' 

2325672 

50.        7289405287 

47. 

4502360 

51.      185205370000 

48. 

62840285 

52.     6423691450896 

49. 

425026  951 

53.    75894128247625 

EXERCISES    IN     NOTATION. 

15»  To  express  numbers  by  figures. 

Begin  at  the  left  hand  of  the  highest  period,  and  write 
the  figures  of  each  period  as  though  it  stood  alone. 

If  any  intervening  order,  or  period  is  omitted  in  the 
given  number,  write  ciphers  in  its  place. 

Write  the  following  numbers  in  figures  upon  the  slate 
or  black-board. 

1.  Sixteen,  seventeen,  eighteen,  nineteen,  twenty. 

2.  Twenty-three,  twenty-five,  thirty,  thirty-three. 

3.  Forty-nine,  fifty-one,  sixty,  seventy-four. 

4.  Eighty-six,  ninety-three,  ninety-seven,  a  hundred. 

QUEST. — 15.  How  are  numbers  expressed  by  figures?    If  any  Intervening 
order  ia  omitted  in  the  example,  how  is  its  place  supplied? 


ART.  15.]  NUMERATION.  15 


5.  One  hundred  and  ten. 

r>.  Two  hundred  and  thirty-five. 

7.  Three  hundred  and  sixty. 

8.  Two  hundred  and  seven. 

9.  Four  hundred  and  eighty-one. 

10.  Six  hundred  and  ninety-seven. 

11.  One  thousand,  two  hundred  and  sixty-three. 

12.  Four  thousand,  seven  hundred  and  ninety-nine. 

13.  Sixty-five  thousand  and  three  hundred. 

14.  One   hundred  and  twelve  thousand,  six  hundred 
and  seventy-three. 

]  5.  Three  Imndred  and  forty  thousand,  four  hundred 
and  eighty-five. 

16.  Two  millions,  five  hundred  and  sixty  thousand. 

17.  Eight  millions^  two  hundred   and   five   thousand, 
three  hundred  and  forty-five. 

18.  Ten  millions,  five  hundred  thousand,  six  hundred 
and  ninety-five. 

19.  Seventeen    millions,    six    hundred    and    forty-five 
thousand,  two  hundred  and  six. 

20.  Forty-one  millions,  six  hundred  and  twenty  thou- 
sand, one  hundred  and  twenty-six. 

21.  Twenty-two   millions,  six  hundred   thousand,  one 
hundred  and  forty-seven. 

22.  Three  hundred  and    sixty  millions,  nine  hundred 
and  fifty  thousand,  two  hundred  and  seventy. 

23.  Five  billions,  six  hundred  and  twenty-one  millions, 
seven  hundred  and  forty-seven  thousand,  nine  hundred 
and  fifty- four. 

24.  Thirty-seven  trillions,  four  hundred  and  sixty-three 
billions,  two  hundred  and  ninety-four  thousand,  five  hun- 
dred and  seventy-two. 


ADDITION.  [SECT.  IT. 

SECTION    II. 

ADDITION. 

ART.  16«  Ex.  1.  Henry  paid  4  shillings  for  a  pair  of 
gloves,  7  shillings  for  a  cap,  and  2  shillings  for  a  knife : 
how  many  shillings  did  he  pay  for  all  ? 

Solution. — 4  shillings  and  7  shillings  are  11  shillings, 
and  2  shillings  are  13  shillings.  He  therefore  paid  13 
shillings  for  all. 

CBS.  The  preceding  operation  consists  in  finding  a  single  num- 
ber which  is  equal  to  the  severed  given  numbers  united  together, 
and  is,  called  Addition.  Hence, 

17»  ADDITION  is  the  process  of  uniting  two  or  more 
numbers  in  one  sum. 

The  answer,  or  number  obtained  by  addition,  is  called 
the  sum  or  amount. 

OBS.  When  the  numbers  to  be  added  are  all  of  the  same  kind,  or 
denomination,  the  operation  is  called  Simple  Addition. 

18.  Sign  of  Addition  (  +  ).  The  sign  of  addition  is 
a  perpendicular  crocs  (  +  ),  called  plus,  and  shows  that 
the  numbers  between  which  it  is  placed,  are  to  be  added 
together.  Thus,  the  expression  6  +  8,  signifies  that  6  is 
to  be  added  to  8.  It  is  read,  "  6  plus  8,"  or  "6  added  to  8." 

Note. — The  term  plus,  is  a  Latin  word,  originally  signifying 
"  more."  In  Arithmetic,  it  means  "  added  to." 


QITKST.— 17.  What  is  addition?  What  is  the  answer  called?  Obs.  WJ-.en 
tho  numbers  to  be  added  are  all  of  the  same  denomination,  whnt  is  ihe  «>}»• 
ration  called?  18.  What  is  the  sii?n  of  addition  ?  What  does  it  show  f  Jv'oie. 
What  is  Uio  meaning  of  the  word  plus? 


ARTS.  16—19.] 


ADDITION. 


19»  Sign  of  Equality  (=).  The  sign  of  equality  is 
two  horizontal  lines  (=),  and  shows  that  the  numbers  be- 
tween which  it  is  placed,  are  equal  to  each  other.  Thus, 
the  expression  4  +  3  =  7,  denotes  that  4  added  to  3  are 
equal  to  7.  It  is  read,  "  4  plus  3  equal  7,"  or  "  the  sum 
of  4  plus  3  is  equal  to  7."  18  +  5=7  +  16. 

ADDITION  TABLE. 


2  and 

3  and 

4  and 

5  and 

1  are   3 

1  are   4 

1  are   5 

1  are   6 

2   "    4 

2   "    5 

2   "    6 

2   "    7 

3   "    5 

3   "    6 

3   «    7 

3   "    8 

4   "    6 

4   "   7 

4   "    8 

4   "    9 

5   "    7 

5   "    8 

5   "    9 

5   "   10 

6   "-   8 

6   "    9 

6   "   10 

6   "   11 

7   "   9 

7   "   10 

7   "   11 

7   "   12 

8   "   10 

8   "   11 

8   "   12 

8   "   13 

9   "   11 

9   "   12 

9   "   13 

9   "   14 

10   "   12 

10   "   13 

10   "   14 

10   "   15 

/ 

6  and 

7  and 

8  and 

9  and 

1  are   7 

1  are   8 

1  are   9 

1  are  10 

2   "   8 

2   "    9 

2   "   10 

2   "   11 

3   "   9 

3   "   10 

3   "   11 

3   "   12 

4   "   10 

4   "   11 

4   "   12 

4   "   13 

5   "   11 

5   "   12 

5   "   13 

5   "   14 

6   "   12 

6   "   13 

6   M   14 

6   "   15 

7   "   13 

7   "   14 

•  7   "   15 

7   "   16 

8   "   14 

8   "   15 

8   "   16 

8   "   17 

9   "   15 

9   "   16 

9   "   17 

9   "   IS 

10   "   16 

10   «   17 

10   "   18 

10   "   19 

Note. — It  te  an  interesting  and  profitable  exercise  for  young  pupils 
to  recite  tables  in  concert.  But  it  will  not  do  to  depend  upon  this 
method  alone.  It  is  indispensable  for  every  scholar  who  desires  to 
»ie  accurate  either  in  arithmetic  or  business,  to  have  the  common 

QTTEST.— Itf.  What  is  the  sign  of  equality?    What  doef  it  ehowf 


18  ADDITION.  [SECT.  II 

arithmetical  tables  distinctly  and  indelibly  fixed  in  his  mind.  Hence 
after  a  table  has  been  repeated  by  the  class  in  concert,  or  individ- 
ually, the  teacher  should  ask  many  promiscuous  questions,  to  prevent 
its  being  recited  mechanically,  from  a  knowledge  of  the  regular  in- 
crease of  numbers. 

EXAMPLES. 

2O«    When  the  sum  of  a  column  does  not  exceed  9. 

Ex.  1.  George  gave  37  cents  for  his  Arithmetic,  and 
42  cents  for  his  Reader :  how  many  cents  did  he  give  for 
both? 

Directions. — Write  the  numbers          Operation. 
under   each   other,  so   that  units       OT-  ^ 
may  stand  under  units,  tens  under       jj  g 
tens,  and  draw  a  line  beneath  them.       3    7  price  of  AritK 
Then,  beginning  at  the  right  hand       ±   %     «     of  j{eaj 

or  units,  add  each  column  sepa-      

rately  in  the  following  manner : —       7   9  Ans. 

2  units  and  7  units  are  9  units.      Write  the  9  in  units' 

place  under  the  column  added.     4  tens  and  3  tens  are 

7  tens.     Write  the  7  hi  tens'  place.     The  amount  is  79 

cents. 

Write  the  following  examples  upon  the  slate  or  black- 
board, and  find  the  sum  of  each  in  a  similar  manner : 

(2.)  (3.)  (4.)  (5.) 

26  231  623  5734 

42  358  145  4253 

(6.)  (7.)  (8.) 

425  3021  5120 

132  1604  2403 

321  2142  1375 

10.  What  is  the  sum  of  4321  and  2475  ? 
11*  What  is  the  sum  of  32562  and  56214  ? 
12.  What  is  the  sum  of  521063  and  465725  ? 


ARTS.  20 — 22.  J  ADDITION.  19 

21.  When  the  sum  of  a  column  exceeds  9. 

13.  A  merchant  sold  a  quantity  of  flour  for  458  dollars, 
a  quantity  of  tea  for  887  dollars,  and  sugar  for  689  dol- 
lars :  how  much  did  he  receive  for  all  ? 

Having  written  the  numbers  as  Operation. 

before,  we  proceed  thus:  9  units       458  price  of  flour, 
and  7  units  are  16  units,  and  8       887     "      of  tea. 
are  24  units,  or  we  may  simply       689     "      of  sugar, 
say  9  and  7  are  16,  and  8  are  24.     2034  dollars.  Ans. 
Now  24  is  equal  to  2  tens  and 

4  units.  We  therefore  set  the  4  units  or  right  hand  figure 
in  units'  place,  because  they  are  units  ;  and  reserving  the 
2  tens  or  left  hand  figure  in  the  mind,  add  it  to  the  column 
of  tens  because  it  is  tens.  Thus,  2  (which  was  reserved) 
and  8  are  10,  and  8  are  18,  and  5  are  23.  Set  the  3  or 
right  hand  figure  under  the  column  added,  and  reserving 
the  2  or  left  hand  figure  in  the  mind,  add  it  to  the  column 
of  hundreds,  because  it  is  hundreds.  Thus,  2  (which  was 
reserved)  and  6  are  8,  and  8  are  16,  and  4  are  20.  Set 
the  0  or  right  hand  figure  under  the  column  added  ;  and 
since  there  is  no  other  column  to  be  added,  write  the  2 
in  thousands'  place,  because  it  is  thousands. 

N.  B.  The  pupil  must  remember,  in  all  cases,  to  set  down  the 
whole  sum  of  the  last  or  left  hand  column. 

22.  The  process  of  reserving  the  tens  or  left  hand  fig- 
ure, when  the  sum  of  a  column  exceeds  9,  and  adding  it 
mentally  to  the  next  column,  is  called  carrying  tens. 

Find  the  sum  of  each  of  the  following  examples  in  a 
similar  manner : 

(14.)  (15.)  (16.)  (17.) 

856  364  6502  8245 

764  488  497  4678 

1020  Ans.  602  8301  362 


$0  ADDITION.  [SECT.  II 

23*  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL  RULE  FOR  ADDITION. 

I.  Write  the  numbers  to  be  added  under  each  other,  so 
that  units  may  stand  under  units,  tens  under  tens,  &c. 

II.  Beginning  at  the  right  hand,  add  each  column  sepa- 
rately, and  if  the  sum  of  a  column  does  not  exceed  9,  write 
it  under  the  column  added.     But  if  the  sum  of  a  cciumn 
exceeds  9,  write  the  units'  figure  under  the  column  and 
carry  the  tens  to  the  next  column. 

III.  Proceed  in  this  manner  through  all  the  orders,  and 
finally  set  down  the  whole  sum  of  the  last  or  left  hand 
column. 

24.  PROOF. — Beginning  at  the  top,  add  each  column 
downward,  and  if  the  second  result  is  the  same  as  the 
first,  the  work  is  supposed  to  be  right. 

EXAMPLES  FOR  PRACTICE. 

(1.)  (2.)  (3.)  (4.) 

Pounds.  Feet.  Dollars.  Yards. 

25        113        342       4008 

46        84       720       635 

_84       2V6       898        43 

(5.)  (6.)  (7.)  (8.) 

684  336  6387  8261 

948  859  593  387 

569  698  3045  13 

203  872  15  7 

9.  What  is  the  sum  of  46  inches  and  38  inches  f 


QUKST. — 23.  How  do  you  write  numbers  for  addition?  When  the  mini  of  a 
•olumn  does  not  exceed  9,  how  proceed?  When  it  exceeds  9,  how  proceed? 
2*2.  What  is  meant  by  carrying  the  tens  ?  What  do  you  do  with  the  sum  of 
I  he  last  co  h;  inn  ?  24.  How  is  addition  proved  ? 


AK.TS.  23,  24.]  ADDITION.  21 

10.  What  is  the  sum  of  51  feet  and  63  feet  ? 

11.  What  is  the  sum  of  75  dollars  and  93  dollars? 

12.  Add  together  45  rods,  63  rods,  and  84  rods. 

13.  Add  together  125  pounds,  231  pounds,  426  pounds. 
J.4.__Add  together  267  yards,  488  yards,  and  625  yards. 

15.  Henry  traveled  256  miles  by  steamboat  and  320 
miles  by  Railroad  :  how  many  miles  did  he  travel  ? 

16.  George  met  two  droves  of  sheep ;  one  contained 
461,  and  the  other  375 :  how  many  sheep  were  there  in 
both  droves  ? 

17.  If  I  pay  230  dollars  for  a  horse,  and  385  dollars  for 
a  chaise,  how  much  shall  I  pay  for  both  ? 

18.  A  farmer  paid  85  dollars  for  a  yoke  of  oxen,  27 
dollars  for  a  cow,  and  69  dollars  for  a  horse:  how  much 
did  he  pay  for  all  ? 

^lO^Find  the  sum  of  425,  346,  and  68^ 

20.  Find  the  sum  of  135,  342,  and  778. 

21.  Find  the  sum  of  460,  845,  and  576. 

22.  Find  the  sum  of  2345,  4088,  and  260. 

23.  Find  the  sum  of  8990,  5632,  and  5863. 
^24^ Find  the  sum  of  2842,  6361,  and  523. 

25.  Find  the  sum  of  602,  173,  586,  and  408. 

26.  Find  the  sum  of  424,  375,  626,  and  75. 

27.  Find  the  sum  of  24367,  61545,  and  20372. 

28.  Find  the  sum  of  43200,  72134,  and  56324. 

29.  A  young  man  paid  5  dollars  for  a  hat;  6  dollars 
for  a  pair  of  boots,  27  dollars  for  a  suit  of  clothes,  and  19 
dollars  for  a  cloak :  how  much  did  he  pay  for  all  ? 

jv  30.  A  man  paid  14  dollars  for  wood,  16  dollars  for  a 
slave,  and  28  dollars  for  coal :  how  many  dollars  did  he 
pay  for  all  ? 

31.  A  farmer  bought  a  plough  for  13  dollars,  a  cart 
for  46  dollars,  and  a  wagon  for  61  dollars :  what  was  the 
price  of  all  ? 


22  ADDITION.  [SECT.  H. 

32.  What  is  the  sum  of  261+31+256  +  17  ? 

33.  What  is  the  sum  of  163+478  +  82  +  19  ? 
34.'  What  is  the  sum  of  428  +  632  +  76  +  394  ? 
"35.  What  is  the  sum  of  320  +  856  +  100  +  503  ? 

36.  What  is  the  sum  of  641+108  +  138  +  710  ? 

37.  What  is  the  sum  of  700  +  66  +  970  +  21  ? 

38.  What  is  the  sum  of  304+971+608+496  ? 
_39.  What  is  the  sum  of  848  +  683+420  +  668?^, 

40.  What  is  the  sum  of  868+45  +  17  +  25  +  27+38? 

41.  What  is  the  sum  of  641  +  85+580+42  +  7+63? 

42.  What  is  the  sum  of  29  +  281+7+43  +  785+46  ? 

43.  A  farmer  sold  25  bushels  of  apples  to  one  man,  17 
bushels  to  another,  45  bushels  to  another,  and  63  bushels 
to  another :  how  many  bushels  did  he  sell  ? 

44.  A  merchant  bought  one  piece  of  cloth  containing 
25  yards,  another  28  yards,  another  34  yards,  and  an- 
other 46  yards :  how  many  yards  did  he  buy? 

"^45.  A  man  bought  3  farms;  one  contained  120  acres, 
another  246  acres,  and  the  other  365  acres :  how  many 
acres  did  they  all  contain? 

46.  A  traveler  met  four  droves  of  cattle;  the  first  con- 
traned  260,  the  second  175,  the  third  342,  and  the  fourth 
420 :  how  many  cattle  did  the  four  droves  contain  ? 

47.  A  carpenter  built  one  house  for  2365  dollars,  an- 
other for  1648  dollars,  another  for  3281  dollars,  and  an- 
other for  5260  dollars:  how  much  did  he  receive  for  all? 

48.  Find  the  sum  of  six  hundred  and  fifty-four,  eighty- 
nine,  four  hundred  and  sixty- three,  and  seventy-six. 

49.  Find  the  sum  of  two  thousand  and  forty-seven, 
three  hundred  and  forty-five,  thirty-six,  and  one  hundred. 

50.  In  January  there  are  31  days,  February  28,  March 
31,  April  30,  May  31,  June  30,  July  31,  August  31,  Sep- 
tember 30,  October  31,  November  30,  and  December  31 : 
how  many  days  are  there  in  a  year  ? 


ART.  24.a.] 


ADDITION. 


24.a.  Accuracy  and  rapidity  in  adding  can  be  ac- 
quired only  by  practice.  The  following  exercises  are  de- 
signed to  secure  this  important  object. 

OBS.  1.  In  solving  the  following  examples,  it  is  recommended 
to  the  pupil  simply  to  pronounce  the  result,  as  he  adds  each  suc- 
cessive figure.  Thus,  in  Ex.  1,  instead  of  saying  2  and  2  are  4, 
and  2  are  6,  &c.,  proceed  in  the  following  manner :  "  two.  four,  six, 
eight,  ten,  twelve,  fourteen,  sixteen,  eighteen,  twenty."  Set  down 
naught  and  carry  two.  "  Two,  (to  carry)  three,  six,  nine."  &c. 

2.  When  two  or  three  figures  taken  together  make  10,  as  8  and  2, 
7  and  3.  or  2,  3,  and  5,  it  accelerates  the  process  to  add  their  sura 
at  once.  Thus,  in  Ex.  4,  the  pupil  should  say:  "  ten  (1+9),  six- 
teen (6),  twenty-six  (5+5).  thirty-six  (2+8),"  &c. 


(1.) 
32 

32 
32 
32 
32 
32 
32 
32 
32 

12. 

(5.) 
614 
452 
528 
539 
420 
385 
355 
134 
976 
468 


(2.) 
654 
654 
654 
654 
654 
654 
654 
654 
654 
114 

(6.) 
2140 
8963 
1232 
7855 
2123 
3333 
7674 
4521 
6589 
2637 


(3.) 
987 
987 
987 
987 
987 
987 
987 
987 
987 
117 


8675 
2433 
6182 
2921 
2209 
4863 
6558 
5434 
5276 
8789 


463 
647 
455 
258 
572 
595 
615 
346 
729 

m 

(8.) 
9244 
1432 
7234 
2523 
8440 
4346 
6704 
1852 
9258 
8106  * 


24 


ADDITION. 


[SECT.  IL 


(9.) 
4360 

(10.) 
9201 

(11.) 

42671 

(12.) 

62125 

7046 

7283 

68439 

31684 

5724 

4627 

32074 

22435 

5385 

6436 

47616 

16725 

8275 

9874 

30045 

94381 

9342 

8400 

26765 

25036 

6768 

6645 

10850 

85474 

5020 

4365 

25232 

10325 

9384 

8640 

43679 

42312 

(13.) 

2720 

(14.) 
5764 

(15.) 

27856 

(16.) 
47639 

4382 

5346 

32534 

23421 

2640 

3042 

20631 

34323 

3047 

5268 

34327 

71036 

2163 

3161 

53102 

62342 

6741 

2560 

92763 

57654 

1360 

7304 

51834 

32103 

7056 

2723 

23452 

53728 

3554 

8459 

62327 

61342 

4275 

6715 

^    50632 

23201 

(17.) 
4521 

(18.) 
6845 

(19.) 
75360 

(20.) 
89537 

3432 

3151 

27838 

23264 

4327 

2327 

42627 

41728 

6238 

4235 

34872 

74263 

5494 

2835 

63538 

21031 

3217 

5473 

54321 

53426 

2382 

9864 

63054 

91342 

4723 

3103 

29872 

23465 

3604 

7382 

63541 

38754 

2352 

5461 

53279 

94642 

T.  24.«.J  ADDITION.  25 


/O"l  \ 

(22.) 

(23.) 

(24.) 

8564 

56,430 

84,703 

341,725 

4736 

31,932 

19,384 

227,265 

3405 

29,754 

21,705 

311,265 

5037 

46,536 

.43,641 

200,378 

6571 

86,075 

27,469 

421,850 

7439 

30,235 

52,267 

370,432 

4525 

41,623 

61,383 

174,370 

3137 

45,810 

75,604 

831,031 

2743 

56,239^ 

43,876 

580,456 

(25.) 

^T 
(26.) 

^  (27.) 

(28.) 

7243 

31,625 

\  68,901 

460,732 

2034 

51,482 

50,345 

804,045 

3710 

49,061 

75,005 

346,325 

5634 

80,604 

29,450 

450,673 

1730 

24,540 

80,063 

859,721 

5613 

67,239 

91,700 

236,548 

3005 

24,307 

43,621 

632,462 

7206 

58,392 

47,834 

753,324 

4354 

70,300 

83,276 

970,300 

7821 

56,749 

25,327 

267,436 

(29-) 

(30.) 

x>  (31.) 

(32.) 

6458 

75,340 

64,268 

346,768 

2435 

6,731 

405 

21,380 

4678 

748 

1,708 

4,075 

4962 

68,451 

43,671 

126,849 

5143 

396 

72,049 

257 

8437 

7,503 

492 

1,305 

7643 

46,075 

1,760 

24,350 

6850 

1,290 

25,357 

439,871 

7063 

25,738 

1,434 

40,300 

8324 

46,803 

84,162 

601,734 

26 


ADDITION. 


[SECT.  II. 


(33.) 

(34.) 

(35.) 

(36.) 

423,674 

632,153 

317,232 

412,783 

307,316 

420,432 

203,671 

631,432 

730,248 

323,680 

334,263 

572,316 

506,213 

507,325 

210,600 

231,254 

110,897 

383,734 

356,237 

673,323 

206,341 

634,156 

264,871 

217,067 

324,563 

450,071 

531,634 

306,421 

185,174 

808,463 

342,106 

764,315 

364,230 

160,705 

768,342 

207,254 

150,176 

300,430 

407,821 

843,552 

843,204 

461,007 

311,289 

321,634 

370,679 

297,313 

564,735 

502,543 

445,168 

813,792 

470,334 

617,405 

370,432 

200,406 

436,216 

506,032 

5,338,31  5  Ans. 

6,388,667 

Ans.  621,353 

762,573 

(37.) 

(38.) 

(39.) 

(40.) 

674,326 

783,457 

863,725 

958,439 

453,403 

675,306 

755,387 

843,670 

561,734 

858,642 

964,845 

784,561 

789,867 

246,468 

836,450 

976,435 

645,275 

587,649 

645,265 

833,406 

576,182 

523,731 

783,842 

797,624 

934,922 

445,372 

532,653 

845,358 

423,641 

832,148 

647,412 

978,262 

561,232 

465,363 

481,735 

784,643 

143,671 

642,742 

824,364 

865,343 

238,406 

830,423 

537,572 

976,736 

453,762 

256,372 

463,489 

853,974 

984,651 

662,456 

827,343 

467,852 

845,359 

572,834 

642,536 

948,685 

967,423 

864,213 

725,342 

896,872 

ARTS.  25 — 27.]  SUBTRACTION.  27 

SECTION  III. 

SUBTRACTION. 

ART.  25»  Ex.  1.  Charles  having  15  cents,  gave  6  cents 
for  an  orange  :  how  many  cents  did  he  have  left  ? 

Solution. — 6  cents  taken  from  15  cents  leave  9  cents. 
Therefore  he  had  9  cents  left. 

OBS.  The  preceding  operation  consists  in  taking  a  less  number 
from  a  greater,  and  is  called  Subtraction.  Hence, 

26.  SUBTRACTION  is  the  process  of  finding  the  differ- 
ence betioeen  two  numbers. 

The  answer,  or  number  obtained  by  subtraction,  is  called 
the  difference  or  remainder. 

OBS.  1.  The  number  to  be  subtracted  is  often  called  the  suldra 
hend,  and  the  number  from  which  it  is  subtracted,  the  minuend 
These  terms,  however,  are  calculated  to  embarrass,  rather  than 
assist  the  learner,  and  are  properly  falling  into  disuse. 

2.  When  the  given  numbers  are  all  of  the  same  kind,  or  denomi 
nation,  the  operation  is  called  Simple  Subtraction. 

27 •  Sign  of  Subtraction  (  — ).  The  sign  of  subtrac- 
tion is  a  horizontal  line  ( — ),  called  minus,  and  shows 
that  the  number  after  it  is  to  be  subtracted  from  the  one 
before  it.  Thus  the  expression  7 —  3,  signifies  that  3  is  to  be 
subtracted  from  7  ;  and  is  read,  "  7  minus  3,"  or  "  7  less  3." 
Read  the  following :  18  —  7  =  20  —  9.  23  —  10  =  16  —  3. 
35  —  8=31  —  4. 

Note. — The  term  minus  is  a  Latin  word  signifying  less. 

QUEST.— CJG.  What  is  subtraction?  What  is  tho  answer  called?  Cbs. 
What  is  the  number  to  be  subtracted  sometimes  called  ?  That  from  which  \\ 
is  subtracted  ?  When  the  given  numbers  are  of  the  same  denomination,  what 
is  the  operation  called  ?  527.  What  is  the  sign  of  subtraction  ?  What  does  it 
show  1  Note.  What  is  the  meaning  of  the  term  minus? 


28 


SUBTRACTION. 


[SECT.  111. 


SUBTRACTION   TABLE. 


2  from 

3  from 

4  from 

5  from 

2  leaves    0 

3  leaves    0 

4  leaves    0 

5  leaves    0 

3      "        1 

4      "        1 

5       "         1 

6       "         1 

4      "        2 

5      "        2 

6      "        2 

7      "        2 

5      "        3 

6      "        3 

7      "        3 

8       "        3 

6      "        4 

7      "        4 

8      "        4 

9       "        4 

7      "        5 

8      "        5 

9      "        5 

10      "        5 

8      "        6 

9      "        6 

10      "        6 

11       "        6 

9      "        7 

10      "        7 

11       "        7 

12      "        7 

10      "        8 

11      "        8 

12      "        8 

13       "        8 

11      "        9 

12      "        9 

13       "        9 

14      "        9 

12      "      10 

13      "      10 

14      "      10 

15      "      10 

6  from 

7  from 

8  from 

9  from 

6  leaves    0 

7  leaves    0 

8  leaves    0 

9  leaves    0 

7      "        1 

8      "        1 

9      "        1 

10      "        1 

8      "        2 

9      "        2 

10      "        2 

11       "        2 

9      "        3 

10      "        3 

11      "        3 

12      "        3 

10      "        4 

11      "        4 

12      "        4 

13       "        4 

11       "        5 

12      "        5 

13       "        5 

14      "        5 

12      "        6 

13      "6 

14      "        6 

15      "        6 

13      "        7 

14      "        7 

15       "        7 

16      "        7 

14      "        8 

15       "        8 

16      "        8 

17      "        8 

15      "        9 

16      "        9 

17      "        9 

18      "        9 

16      "      10 

17      "      10 

18      "      10 

19       "      10 

OBS.  This  Table  is  the  reverse  of  Addition  Table.  Hence,  if  the 
pupil  has  thoroughly  learned  that,  it  will  cost  him  but  little  time  or 
trouble  to  learn  this.  (See  observations  under  Addition  Table.) 

EXAMPLES. 

2  8  •  When  each  figure  in  the  loiver  number  is  smaller 
than  the  figure  above  it. 

1.  A  farmer  raised  257  bushels  of  apples,  and  123 
bushels  of  pears  :  how  many  more  apples  did  he  raise 
than  pears  ? 


ARTS.  28,  29.  j  SUBTRACTION.  29 

Directions. — Write     the     less  Operation. 

number    under    the  greater,    so 
that  units  may  stand  under  units,  *g    M*  £ 

tens  under  tews,  &c.,  and  draw  a  j2  3    g 

line   beneath   them.     Beginning  257  apples, 

frith  the  units  or  right  hand  fig-  123  pears, 

nre,  subtract  each  figure  in  the      Hem.    134  bush. 
lower  number   from    the    figure 

above  it,  in  the  following  manner :  3  units  from  7  units 
leave  4  units.  Write  the  4  in  units'  place  under  the 
figure  subtracted.  2  tens  from  5  tens  leave  3  tens ;  set 
3  in  tens'  place.  1  hundred  from  2  hundred  leaves  1  hun- 
dred ;  write  the  1  hundred  in  hundreds'  place. 

Solve  the  following  examples  in  a  similar  manner : 

(2.)  (3.)  (4.)  (5.) 

From  45  68  276  698 

Take  _21.  66_  123  453 

(6.)  (7.)  (8.)  (9.) 

From  54  dolls.        76  pounds.        257  yds.       325  shil. 
Take  J23  dolls.        64  pounds.        142  yds.        103  shil. 

10.  Samuel  having  436  marbles,  lost    214  of  them: 
how  many  had  he  left  ? 

29*    When  a  figure  in  the  lower  number  is  larger  than 
the  figure  above  it. 

— ,  11.  A  man  bought  63   bushels  of  wheat,  and  after- 
wards sold  37 :  how  many  bushels  had  he  left? 

It  is  obvious  that  we  cannot  take  7       First  Method. 
units  from  3  units,  for  7  is  larger  than  63 

3  ;  we  therefore  add  10  to  the  3  units,  .  37 

and  it  will  make  13  units  ;  then  7  from      Rem.  26  bu. 
1 3  leaves  6  ;  write  the  6  in  units'  place 
under  the  figure  subtracted.     To  compensate  for  the  10 


30  SUBTRACTION.  [SECT.  Ill 

units  we  added  to  the  upper  figure,  we  add  1  ten  to  the 

3  tens  or  next  figure  in  the  lower  number,  and  it  makes 

4  tens ;  and  4  tens  from  6  tens  leave  2  tens :  write  the  2 
in  tens'  place.  Ans.  26  bushels. 

We  may  also  illustrate  the  process  of  borrowing  in  he 
following  manner : 

63  is  composed  of  6  tens  and  3         Second  Method. 
units.     Taking  1  ten  from  6  tens,          63  =  50  +  13 
and  adding  it  to  the  3  units,  we          37  =  30+  7 
have63  =  50+13.  Separating  the     .Sera.  =  20  +   6,  or  26. 
lower  number  into  tens  and  units, 

we  have  37  =  30  +  7.  Now,  substracting  as  before,  7 
from  13  leaves  6.  Then  as  we  took  1  ten  from  the  6  tens, 
we  have  but  5  tens  left ;  and  3  tens  from  5  tens  leave  2 
tens.  The  remainder  is  26,  the  same  as  before. 

3O.  The  process  of  taking  one  from  a  higher  order  in 
the  upper  number,  and  adding  it  to  the  figure  from  which 
the  subtraction  is  to  be  made,  is  called  borrowing  ten,  and 
is  the  reverse  of  carrying  ten.  (Art.  22.) 

OBS.  When  we  borrow  ten  we  must  always  remember  to  pay  it. 
This  may  be  done,  as  we  have  just  seen,  either  by  adding  1  to  the 
next  figure  in  the  lower  number,  or  by  considering  the  next  figure 
in  the  upper  number  1  less  than  it  is . 

"""  12.  From  240  subtract  134,  and  prove  the  operation. 

Since   4  cannot  be  taken  from  0,  we  Operation. 

borrow  10;  then  4  from  10  leaves  6.     1  240 

added  to  3  (to  compensate  for  the  10  we  134 

borrowed)*  makes  4,  and  4  from  4  leaves  0.  106  Ana, 
1  from  2  leaves  1. 

PROOF. — We  add  the  remainder  Proof. 

to  the  smaller  number,  and  since  the  134  less  No. 

sum  is  equal  to  the  larger  number,  106  remainder, 

the  work  is  right.  240  greater  No. 


ARTS.  30 — 32.]          SUBTRACTION.  31 

Solve  the  following  examples,  and  prove  the  operation 

(13.)  (14.)  (15.)  (16.) 

From  375  5273  6474  8650 

Take_238  2657  3204  5447 

^•17.  From  8461875,  take  3096208. 

31*  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL   RULE   FOR   SUBTRACTION. 

I.  Write  the  less  number  under  the  greater,  so  that  units 
may  stand  under  units,  tens  under  tens,  &c. 

II.  Beginning  at  the  right  hand,  subtract  each  figure  in 
the  lower  number  from  the  figure  above  itt  and  set  the  re- 
mainder under  the  figure  subtracted. 

III.  When  a  figure  in  the  lower  number  is  larger  than 
that  above  it,  add  10  to  the  upper  figure  ;  then  subtract  as 
before,  and  add  1  to  the  next  figure  in  the  lower  number. 

32*  PROOF. — Add  the  remainder  to  the  smaller  num- 
ber ;  and  if  the  sum  is  equal  to  the  larger  number^  the  work 
is  right. 

OBS.  This  method  of  proof  depends  upon  the  obvious  principle, 
that  if  the  difference  between  two  numbers  be  added  to  the  less,  the 
sum  must  be  equal  to  the  greater. 

EXAMPLES    FOR    PRACTICE. 


(1.) 

Fiom  325 

(2.) 
431 

(3.) 
562 

(4.) 

eoo 

Take  108 

249 

320 

231 

(5.) 
From  2230 

(6.) 
3042 

(»•) 

6500 

(8.) 
8435 

k  Take  1201 

2034 

3211 

5001 

QUEST.— 31.  How  do  you  write  numbers  for  siibtraetion  ?  Where  do  you 
begin  to  subtract?  When  a  figure  in  the  lower  Dumber  is  larger  than  the  one 
above  it,  how  do  you  proceed  ?  32.  How  is  bubti-ucl.ou  proved? 


32  SUBTRACTION.  [SECT.    Ill 

(9-)  (10.)  (11.) 

From  45100  826340  1000000 

Take   10000  513683  999999 

12.  From  132  dollars  subtract  109  dollars. 

13.  From  142  bushels  subtract  85  bushels. 

14.  From  375  pounds  subtract  100  pounds. 
35.  From  698  yards  subtract  85  yards. 

16.  From  485  rods  subtract  175  rods. 

17.  Take  230  gallons  from  460  gallons. 

18.  Take  168  hogsheads  from  671  hogsheads. 

19.  Take  192  bushels  from  268  bushels. 
jyxjiom  275  dollars  take  148  dollars. 

21.  From  468  pounds  take  219  pounds. 

22.  From  3246  rods  take  2164  rods. 

23.  From  45216  take  32200. 

24.  From  871410  take  560642. 

25.  From  926500  take  462126. 

26.  From  6284678  take  1040640. 

27.  468— -423.  37.  17265—13167. 

28.  675 — 367.  38.  21480—20372. 

29.  800—560.  39.  30671—26140. 

30.  701—643.  40.  45723—31203. 

31.  963—421.  41.   81647—57025. 

32.  3263—1242.  42.  265328—140300. 

33.  4165—2340.  43.   170643—106340. 

34.  5600—3000.  44.  465746—241680. 

35.  7246—4161.  45.  694270— 59C  395. 

36.  8670—7364.  46.  920486—500000. 

47.  A  man  having  235  sheep,  lost  163  of  them :  how 
many  had  he  left  ? 

48.  A  farmer  having  500  bushels  of  wheat,  sold  278 
bushels :  how  much  wheat  had  he  left  ? 

49.  A  man  paid  625  dollars  for  a  carnage  and  430 


ART.  32.]  SUBTRACTION.  33 

dollars  for  a  span  of  horses :  how  much  more  did  he  pay 
for  his  carriage  than  for  his  horses  ? 

50.  A  man  gave  1263  dollars  for  a  lot,  and  2385  dol- 
lars for  building  a  house :  how  much  more  did  his  house 
cost  than  his  lot? 

51.  If  a  person  has  3290  dollars  in  real  estate,  and 
owes  1631  dollars,  how  much  is  he  worth? 

52.  A  man  gave  his  son  8263  dollars,  and  his  daughter 
5240  dollars :  how  much  more  did  he  give  his  son  than 
his  daughter? 

53.  A  man  bought  a  farm  for  9467  dollars,  and  sold 
it  for  11230  dollars :  how  much  did  he  make  by  his  bar- 
gain ? 

54.  If  a  man's  income  is  10000  dollars  a  year,  and  his 
expenses  6253  dollars,  how  much  will  he  lay  up  ? 

55.  The  captain  of  a  ship  having  a  cargo  of  goods 
worth  29230  dollars,  threw  overboard  in  a  storm  13216 
dollars'  worth :  what  was  the  value  of  the  goods  left  ? 

56.  A  merchant  bought  a  quantity  of  goods  for  12645 
dollars,  and  afterwards  sold   them  for  13960    dollars: 
how  much  did  he  gain  by  his  bargain  ? 

57.  A  man  paid  23645  dollars  for  a  ship  and  after- 
wards sold  it  for  18260  dollars  :  how  much  did  he  lose 
by  his  bargain  ? 

58.  The  salary  of  the  President  of  the  United  States  is 
25000  dollars  a  year  ;  now  if  his  expenses  are  19265  dol- 
lars, how  much  will  he  lay  up  ? 

59.  A  general  before  commencing  a  battle,  had  35260 
loldiers  in  his  army ;  after  the  battle  he  had  only  21316: 
how  many  soldiers  did  he  lose  ? 

60.  The  distance  of  the  sun  from  the  earth  is  95000000 
miles ;  the  distance  of  the  moon  from  the  earth  is  240000 
miles :  how  much  farther  from  the  earth  is  the  sun  than 
the  moon  ? 


31  SUBTRACTION.  [SECT.   III. 

EXAMPLES    INVOLVING    ADDITION    AND    SUBTRACTION. 

61.  Henry  bought  63  oranges  of  one  grocer,  and  26 
of  another;  he  afterwards  sold  72:    how  many  oranges 
did  he  have  left? 

62.  Charles  had  47  marbles,  and  his  father  gave  him 
36  more ;  he  afterwards  lost  50 :  how  many  marbles  did 
he  then  have  ? 

63.  A  farmer  having   158  sheep,  lost  30  of  them  by 
sickness  and  sold  52 :  how  many  sheep  did  he  have  left? 

64.  Sarah's  father  gave  her  60  cents,  and  her  mother 
gave  her  54  cents ;  if  she  spends  62  cents  for  a  pair  of 
gloves,  how  many  cents  will  she  have  left  ? 

65.  A  merchant  purchased  a  piece  of  silk  containing 
78  yards;  he  then  sold  18  yards  to  one  lady,  and  17  to 
another :  how  many  yards  had  he  left  ? 

66.  If  a  man  has   property  in  his  possession  worth 
215   dollars,   and   owes   39   dollars   to  one  person,  and 
54  dollars  to  another,  how  much  money  will  he  have  left, 
when  he  pays  his  debts  ? 

67.  If  a  man's  income  is  185  dollars  per  month,  and 
he  pays  35  dollars  for  house  rent,  and  63  dollars  for  pro- 
visions per  month,  how  many  dollars  will  he  have  left  for 
other  expenses  ? 

68.  George  having  74  pears,  gave  away  43  of  them; 
if  he  should   buy  35   more,  how  many   would  he  then 
have? 

69.  If  you  add  115  to  78,  and  from  the  sum  take  134, 
what  will  the  remainder  be  ? 

70.  If  you  subtract  93  from  147,  and  add  110  to  the 
remainder,  what  will  the  sum  be  ? 

71.  A  merchant  purchased  125  pounds  of  butter  of 
one  dairy-man,  and  187  pounds  of  another;  he  afterwards 
sold  163  pounds:  how  many  pounds  did  he  have  left? 


ART.  32.  J  SUBTRACTION.  85 

72.  A  'miller  "bought  200  bushels  of   wheat  of  one 
farmer,  and  153  bushels  of  another;  he  afterwards  sold 
189  bushels:  how  many  bushels  did  he  have  left? 

73.  A  man  traveled  538  miles  in  3  days  ;  the  first  day 
tie  traveled  149  miles,  the  second  day,  126  miles:  how 
far  did  he  travel  the  third  day  ? 

74.  A  grocer  bought  a  cask  of  oil  containing  256  gal- 
lons ;  'after  selling  93  gallons,  he  perceived  the  cask  was 
leaky,  and  on  measuring  what  was  left,  found  he  had  38 
gallons :  how  many  gallons  had  leaked  out  ? 

75.  A  manufacturer  bought  248  pounds  of  wool  of  one 
customer,  and  361   pounds  of  another;  he  then  worked 
up  430  pounds :  how  many  pounds  had  he  left  ? 

76.  A  man  paid  375  dollars  for  a  span  of  horses,  and 
450  dollars  for  a  carnage ;  he  afterwards  sold  his  horses 
and  carriage  for  1000  dollars;  how  much  did  he  make 
by  his  bargain  ? 

77.  A  grocer  bought  285  pounds  of  lard  of  one  farmer, 
and  327  pounds  of   another;    he  afterwards  sold  110 
pounds  to  one  customer,  and  163  pounds  to  another:  how 
much  lard  did  he  have  left  ? 

78.  A  flour  dealer  having  500  barrels  of  flour  on  hand, 
sold  263  barrels  to  one  customer  and  65  barrels  to  an- 
other :  how  many  barrels  had  he  left  ? 

79.  Harriet  wished  to  read  a  book  through  which  con- 
tained 726  pages,  in  three  weeks  ;  the  first  week  she  read 
165  pages,  and  the  second  week  she  read  264  pages: 
how  many  pages  were  left  for  her  to  read  the  third  week  ? 

80.  A  man  bought  a  house  for  1200  dollars,  and  hav- 
ing laid  out  210  dollars  for  repairs,  sold  it  for  1300  dol- 
lars :  how  much  did  he  lose  by  the  bargain  ? 

81.  A  young  man  having  2000  dollars,  spent  765  the 
first  year  and  843  the  second   year :  how  much  had  he 
left  ? 


30  MULTIPLICATION.  [SECT.  IV 

SECTION  IV. 

MULTIPLICATION. 

ART.  33.  Ex.  1.  What  will  three  lemons  cost,  at  2 
cents  apiece  ? 

Analysis. — Since  1  lemon  costs  2  cents,  3  lemons  will 
cost  3  times  2  cents  ;  and  3  times  2  cents  are  6  cents 
Therefore,  3. lemons,  at  2  cents  apiece,  will  cost  6  cents. 

OBS.  The  preceding  operation  is  a  short  method  of  finding  how 
much  2  cents  will  amount  to,  when  repeated  or  taken  3  times,  and 
is  called  Multiplication.  Thus,  2  cents  -f-  2  cents  -f-  2  cents  are 
6  cents.  Hence, 

34.  MUTIPLICATION  is  theprocess  of  finding  the  amount 
of  a  number  repeated  or  added  to  itself,  a  given  number  of 
times. 

The  number  to  be  repeated  or  multiplied,  is  called  the 
multiplicand. 

The  number  by  which  we  multiply,  is  called  the  mul- 
tiplier, and  shows  how  many  times  the  multiplicand  is  to 
be  repeated  or  taken. 

The  answer,  or  number  produced  by  multiplication,  is 
called  the  product. 

Thus,  when  we  say  5  times  7  are  35,  7  is  tie  multipli- 
cand, 5  the  multiplier,  and  35  the  product. 

OBS.  When  the  multiplicand  denotes  things  of  one  kind,  or  de- 
nomination only,  the  operation  is  called  Simple  Multiplication. 


QUEST. — 34.  What  is  multiplication  ?  What  is  the  number  to  be  -epcated 
or  multiplied  c«il!ed  7  What  (he  number  by  which  we  multiply  T  What  does 
the  multiplier  show  ?  What  is  the  answer  called  ?  When  wo  say  5  times  7 
»re  35,  which  ie  th»  multiplicand?  Which  the  multiplier?  Which  tho 
product?  Obs.  When  the  multiplicand  denotes  things  of  one  denomination 
Oi.iy,  what  is  fh<^  r.p^.iftuou  caiieu  1 


ARTS.  33—36.1 


MULTIPLICATION. 


37 


35.  The  multiplier  and  multiplicand  faken  together, 
are  often  called  factors,  because  they  make  >r  produce  the 
product. 

Note. — The  term/actor,  is  a  Latin  word  which  signifies  an  a  gent , 
a  doer,  or  producer. 

MULTIPLICATION  TABLE. 


2  times 

3  times 

4  times 

5  times 

6  times 

7  times 

1    are    2 

1    are     3 

1    ate     4 

1:  are    5 

1    are     0 

1    are   7 

2      • 

4 

2     "      6 

2     ' 

8 

2     ' 

101  2 

'     12 

2 

14 

3      ' 

(5 

3     «      9 

3     ' 

12 

3     ' 

15    3   ( 

18 

3 

21 

4 

8 

4     «     12 

4     ' 

10 

4 

20    4 

24 

4 

28 

5 

10 

5     "     15 

5     « 

20 

5 

25 

5 

30 

5 

35 

6 

12 

;   »    is 

0     ' 

24 

0 

30 

0 

30 

0 

42 

7 

14 

7     »    2L 

7     < 

28 

7 

35 

7 

42 

7 

49 

8 

10 

8     i:     24 

8    ' 

32 

8 

40 

8 

48 

8 

50 

9 

18 

9     »    27 

9     < 

36 

9 

45 

9 

54 

9 

03 

10 

20 

10     «     30 

10     ' 

40 

10 

50 

10 

00 

10 

70 

11 

22 

11     "     33 

11     < 

44 

11     ' 

55 

11 

00 

11 

77 

12 

24 

12     "     30 

12    < 

48 

12    ' 

00 

12 

72 

12 

84 

8  times 

9  times 

10  times 

11  times 

12  times 

1    t 

L  e    8 

1     are    9 

1     are  10 

1    are  1  1 

1    are  12 

2 

16 

2    "     18 

2     "     20 

2 

22 

2 

'     24 

3 

24 

3     "     27 

3     «    30 

3 

33 

3 

1    36 

4 

32 

4     '      30 

4     "    40 

4 

44 

4 

1    48 

5 

40 

5     '     45 

5     "    50 

5 

55 

5 

<     00 

r>  / 

48 

0     <      54 

0    «     00 

6 

06 

0 

1    7-2 

7 

56 

7    '     03 

7    "     70 

7 

77 

7 

1    84 

8 

04 

8     '      72 

8     "    80 

8 

88 

8 

'     96 

9 

72 

9     !      81 

9     "     90 

9 

99 

9 

'  108 

10 

80 

10     '      90 

10     "  100 

10 

110 

10 

'  120 

11 

88 

1  1     '      99 

11     "  110 

11 

121 

11 

'   132 

12 

90 

12     '    108 

12    «  120 

12 

132 

12 

'  144 

Note. — 1.  It  will  be  perceived  that  the  several  results  of  multif  ly- 
ing by  10,  are  formed  by  adding  a  naught  or  cipher  to  the  figure  that 
is  to  be  multiplied.  Thus,  10  times  2  are  20 ;  10  times  3  are  30.  &c. 

2.  The  results  of  multiplying  by  5,  terminate  in  5  and  0  alter~ 
nately.  Thus,  5  times  1  are  5 ;  5  times  2  are  10;  5  times  3  are  15, 
5  times  4  are  20,  &c. 


Qi'EST. — 35,  What  are  the  multiplicand  and  multiplier  together  called? 
Why  ?    Mite.  What  does  the  term  factor  signiiy  ? 


38  iMULTIPLICATION.  [SECT.  IV, 

3.  The  first  nine  results  of  multiplying  by  11,  are  formed  by  re- 
peating the  figure  to  be  multiplied.     Thus,  11  times  2  are  22;  11 
times  3  are  33,  &c. 

4.  In  the  successive  results  of  multiplying  by  9.  the  right  hand 
figure  regularly  decreases  by  1,  and  the  left  hand  figure  regularly 
increases  by  1.     Thus,  9  times  2  are  18 ;  9  times  3  are  27  ;  9  time* 
4  are  36,  &c. 

36.  Multiplying  by  1,  is  taking  the  multiplicand  once : 
thus,  4  multiplied  by  1=4. 

Multiplying  by  2,  is  taking  the  multiplicand  twice: 
thus,  2  times  4,  or  4  +  4  =  8. 

Multiplying  by  3,  is  taking  the  multiplicand  three  times: 
thus,  3  times  4,  or  4+4+4  =  12,  &c.  Hence, 

Multiplying  by  any  whole  number,  is  talcing  the  multi- 
plicand as  many  times,  as  there  are  units  in  the  multiplier. 

Note. — The  application  of  this  principle  to  fractional  multipliers 
will  be  illustrated  under  fractions. 

OBS.  From  the  definition  of  multiplication,  it  is  manifest  that  the 
product  is  of  the  same  kind  or  denomination  as  the  multiplicand ; 
br.  repeating  a  number  or  quantity  does  not  alter  its  nature.  Thus, 
if  we  repeat  dollars,  they  are  still  dollars ;  if  we  repeat  yards,  they 
are  still  yards ;  if  we  repeat  an  abstract  number,  that  is,  a  number 
which  does  not  express  any  sensible  object,  the  product  will  be  an 
abstract  number,  &c. 

37*  Sign  of  Multiplication  (X).  The  sign  of  multi- 
plication is  an  oblique  cross  (x),  and  shows  that  the  nnm 
bers  between  which  it  is  placed,  are  to  be  multiplied 
together.  Thus  the  expression  5X3",  signifies  that  5  and 
3  are  to  be  multiplied  together,  and  is  read,  "  5  multiplied 
by  3,"  or  simply  "  5  into  3." 

QUEST.— 36.  What  is  it  to  multiply  by  1  ?  By  2  ?  By  3  ?  What  is  it  to 
nultiply  by  any  whole  number  ?  Obs.  Of  what  kind  or  denomination 

IB  the  product  ?  Why  ?  37.  What  is  the  sign  of  multiplication  ?  Wh:it  does 
it  show?  How  is  the  expression  9x6.  read?  How  6x7=42?  HH.  Does  it 
make  any  difference  in  the  product,  which  factor  is  taken  for  the  multiplier  1 
Illustrate  this  principle  upon  the  blackin«!ni. 


ARTS.  36 — 39.]          MULTIPLICATION.  39 

38»   The  product  of  any  two  numbers  will  be  the  same, 
whichever  factor  is  taken  for  the  multiplier.     Thus, 
If  a  garden  contains  3   rows  of  trees  as  4 

represented   by  the  number  of  horizontal         *.„_       * 
*    ,       •    ,i  .          ,        ,  ***** 

rows  ot  stars  in  the  margin,  and  each  row 

***** 
has  5  trees  as  represented  by  the  number  of 

stars  in  a  row,  it  is  evident,  that  the  whole 
number  of  trees  in  the  garden  is  equal  either  to  the  numb<-t 
of  stars  in  a  horizontal  row,  taken  three  times,  or  to  the 
number  of  stars  in  a  perpendicular  row  taken  Jive  times ; 
that  is,  Aqua!  to  5  X  3,  or  3X5. 

EXAMPLES. 

3Q0   When  the  multiplier  contains  but  ORE  figure. 

Ex.  1.  What  will  3  horses  cost,  at  123  dollars  apiece? 

Analysis. —  Since  1  horse  costs  123  dollars,  3  horses 
will  cost  3  times  123  dollars. 

Directions. — Write  the  multi-  Operation, 

plrer   under   the   multiplicand;  123  multiplicand, 

then,    beginning    at    the    right  3  multiplier.  "  • 

hand,  multiply  each  figure  of  the 

u-  v       a    i     *i  if-   r       Dolls.  369  product. 

multiplicand    by  the  multiplier. 

Thus,  3  times  3  units  are  9  units,  or  we  may  simply  say 
3  times  3  are  9  ;  set  the  9  in  units'  place  under  the  figure 
multiplied.  3  times  2  are  6  ;  set  the  6  in  tens'  place. 
3  times  1  are  3  ;  set  the  3  in  hundreds'  place. 

Note. — The  pupil  should  be  required  to  analyze  every  example, 
and  to  give  the  reasoning  in  full ;  otherwise  the  operation  is  liable 
to  become  mere  guess-work,  and  a  habit  is  formed,  which  is  alike 
destructive  to  mental  discipline  and  all  substantial  improvement. 

Solve  the  following  examples  in  a  similar  manner : 
(2.)  (3.)  (4.)  (5.) 

Multiplicand     34  312  2021  1110 

Multiplier  2  3  4  5 


40  MULTIPLICATION.  [SfiCT.   IV 

(6.)  (7.)  (8.)  (9.) 

Multiplicand,  4022  6102  7110  8101 

Multiplier,        __3  _4  5  __7 

10.  What  will  6  cows  cost  at  23  dollars  apiece. 

Suggestion. — In  this  example  the  product  of  the  differ- 
ent figures  of  the  multiplicand  into  the  multiplier,  exceeds 
9 ;  we  must  therefore  write  the  units'  figure  under  the 
figure  multiplied,  and  carry  the  tens  to  the  next  product 
on  the  left,  as  in  addition.    Thus,  begin- 
ning at  the  right  hand  as  before,  6  times         Operation.. 
3  units  are  18  units,  or  we  may  simply  23  dolls, 

say  6  times  3  are  18.     Now  it  requires  6 

two  figures  to  express  18  ;  we  there-  Ans.  138  dollars- 
fore  set  the  8  under  the  figure  multi- 
plied, and  reserving  the  1,  carry  it  to  the  product  of  the 
next  figure,  as  in  addition.  (Art.  23.)  Next,  6  times  2 
are  12,  and  1  (to  carry)  makes  13.  Since  there  are  no 
more  figures  to  be  multiplied,  we  set  down  the  1 3  in  full. 
The  product  is  138  dollars.  Hence, 

4O.  When  the  multiplier  contains  but  one  figure. 
.  Write  the  multiplier  under  the  multiplicand,  units  un- 
der units,  and  draw  a  line  beneath  them. 

Begin  with  the  units,  and  multiply  each  figure  of  the 
multiplicand  by  the  multiplier,  setting  down  the  result  a)td 
carrying  as  in  addition.  (Art.  23.) 

Multiply  the  following  numbers  together. 

11.  78X4.  18.   524X0. 

12.  96X5.  19.  360X7. 

13.  83X3.  20.  475X4. 

14.  120X7.  21.   792X5. 

15.  138X6.  22.  820X8. 

16.  163X5.  23.   804X7. 

17.  281X8.  24.  968X9. 


ArtTS.  40,  41.]  MULTIPLICATION.  4J 

25.  What  will  175  barrels  of  flour  cost,  at  6  dollars 
per  barrel  ? 

26.  A  man  bought  460  pair  of  boots,  at  5  dollars  a  pair : 
how  much  did  he  pay  for  the  whole  ? 

27.  What  cost  196  acres  of  land,  at  7  dollars  per  acre? 

28.  What  cost  340  ploughs,  at  8  dollars  apiece? 

29.  What  cost  691  hats,  at  7  dollars  apiece? 

30.  What  cost  865  heifers,  at  9  dollars  per  head  ? 

31.  What  cost  968  cheeses,  at  8  dollars  apiece? 

32.  What  cost  1260*sheep,  at  7  dollars  per  head? 

33.  What  cost  9  farms,  at  2365  dollars  apiece  ? 

4 1  •    When  the  multiplier  contains  more  than  ONE  figure. 

34.  A  man  sold  23  sleighs,  at  54  dollars  apiece:  how 
much  did  he  receive  for  them  all  ? 

Suggestion. — Reasoning  as  before,    if   1   sleigh  costs 
54  dollars,  23  sleighs  will  cost  23  times  as  much. 

Directions. — As  it  is  not  Operation. 

convenient  to  multiply  by  23  54  Multiplicand, 

at  once,  we  first  multiply  by  23  Multiplier, 

the  3  units,  then  by  the  2  162  cost  of  3  s. 

tens,  and  add  the  two  results  108       "     "20  s. 

together.     Thus,  3  times  4      Dolls.  1242     "     "  23  s. 
are  12,  set  the  2  under  the 

figure  3,  by  which  we  are  multiplying,  and  carry  the  1 
as  above.  3  times  5  are  15,  and  1  (to  carry)  makes  16. 
Next,  we  multiply  by  the  2  tens  thus :  20  times  4  units 
are  80  units  or  8  tens ;  or  we  may  simply  say  2  times  4 
are  8.  Set  the  8  under  the  figure  2  by  which  we  are 
multiplying,  that  is,  in  tens'  place,  because  it  is  tens. 
2  times  5  are  10.  Finally,  adding  these  two  products 
together  as  they  stand,  units  to  units,  tens  to  tens,  &e., 
we  have  1242  dollars,  which  is  the  whole  product  re- 
quired. 


42  MULTIPLICATION.  [SECT.  IV 

Note. — When  the  multiplier  contains  more  than  one  figure,  the 
several  products  of  the  multiplicand  into  the  separate  figures  of  the 
multiplier,  are  called  partial  products. 

35.  Multiply  45  by  36,  and  prove  the  operation. 

Operation. 

Beginning  at  the  right  hand,  we  45,  Multiplicand, 

proceed  thus:    6  times  5  are  30 ;  36  Multiplier, 

set  the  0  under  the  figure  by  which         270 
we  are  multiplying  ;  6  times  ,4  are       135 
24  and  3  (to  carry)  are  27,  &c.     *       1620  Prod. 

Proof. 

PROOF. — We  multiply  the  mul-  36 

tiplier  by  the  multiplicand,    and  45 

since  the  result  thus  obtained  is         180 
the  same  as   the  product  above,        144 
the  work  is  right.  1620  Prod. 

36.  What  is  the  product  of  234  multiplied  by  165  ? 

Operation. 

Suggestion. — Proceed  in  the  same  man-  234 

ner  as  when  the  multiplier  contains  but  165 

two   figures,    remembering   to  place  the  1170 

right  hand  figure  of  each  partial  product  1404 

directly  under  tke  figure  by  which  you  234 

multiply.  3P610  Ans 

37.  What  is  the  product  of  326  multiplied  by  205  ? 

Suggestion. — Since   multiplying   by  a  Operation. 

cipher  produces  nothing,  in  the  operation  326 

we  omit  the  0  in  the  multiplier.     Thus,  205 

having  multiplied  by  the  5  units,  we  next  1630 

multiply  by  the  2  hundreds,  and  place  the  652 

first  figure  of  this  partial  product  under  66830  Ans. 
the  figure  by  which  we  are  multiplying. 


ARTS.  42,  43.]  MULTIPLICATION.  43 

4:!2»  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL   RULE   FOR   MULTIPLICATION. 

I.  Write  the  multiplier  under  the  multiplicand,  unit* 
under  units,  tens  under  tens,  &c. 

II.  When  the  multiplier  contains  but  ONE  figure,  begin 
ivith  the  units,  and  multiply  each  figure  of  the  multipli- 
cand by  the  multiplier,  setting  down  the  result  and  carry- 
ing as  in  addition.     (Art.  23.) 

III.  If  the  multiplier  contains  MORE   than   one  figure, 
multiply  each  figure  of  the  multiplicand  by  each  figure 
of  the  multiplier  separately,  and  write  the  first  figure  of 
each  partial  product  under  the  figure  by  which  you  are 
multiplying. 

Finally,  add  the  several  partial  products  together^  and 
the  sum  will  be  the  whole  product,  or  answer  required. 

4:3 •  PROOF. — Multiply  the  multiplier  by  the  multipli- 
cand,  and  if  the  second  result  is  the  same  as  the  first,  the 
work  is  right. 

OBS.  1.  It  is  immaterial  as  to  the  result  which  of  the  factors  u 
taken  for  the  multiplier.  (Art.  38.)  But  it  is  more  convenient  an<> 
therefore  customary  to  place  the  larger  number  for  the  multipli- 
cand and  the  smaller  for  the  multiplier.  Thus,  it  is  easier  to  mul- 
tiply 254672381  by  7,  than  it  is  to  multiply  7  by  254672381,  but 
the  product  will  be  the  same. 

2.  Multiplication  may  also  be  proved  by  division,  and  by  casting 
out  the  nines  ;  but  neither  of  these  methods  can  be  explained  here 
without  anticipating  principles  belonging  to  division,  with  which 
the  learner  is  supposed  as  yet  to  be  unacquainted. 


QUEST.— 42.  How  do  you  write  numbers  for  multiplication?  When  the 
multiplier  contains  but  one  figure,  hjw  do  you  proceed?  When  the  multi- 
plier contains  more  than  one  figure,  how  proceed?  41.  Note.  What  is  meant 
by  partial  products?  Whut  is  to  be  done  with  the  partial  products?  43 
How  is  multiplication  proved  V 


44  MULTIPLICATION.  [SECT.  IV. 

EXAMPLES    FOR    PRACTICE. 

1.  Multiply  63  by  4.  10.  Multiply  46  by  10. 

2.  Multiply  78  by  5.  11.  Multiply  52  by  11. 

3.  Multiply  81  by  7.  12.  Multiply  68  by  12. 

4.  Multiply  97  by  6.  13.  Multiply  84  by  13. 

5.  Multiply  120  by  7.  14.  Multiply  78  by  15. 

6.  Multiply  231  by  5.  15.  Multiply  95  by  23. 

7.  Multiply  446  by  8.  16.  Multiply  129  by  35. 

8.  Multiply  307  by  9.  17.  Multiply  293  by  42. 

9.  Multiply  560  by  7.  18.  Multiply  461  by  55. 

19.  If  1   barrel  of  flour  costs  9   dollars,    how  much 
will  38  barrels  cost  ? 

20.  If  1   apple-tree  bears  14  bushels  of  apples,  how 
many  bushels  will  24  trees  bear? 

21.  In  1  foot  there  are  12  inches:  how  many  inches 
are  there  in  28  feet? 

22.  In  1  pound  there  are  20  shillings :  how  many  shil- 
lings are  there  in  31  pounds  ? 

23.  What  will  17  cows  cost,  at  23  dollars  apiece  ? 

24.  What  will  25  tons  of  hay  cost,  at  19  dollars  per  ton  ? 

25.  What  will  37  sleighs  cost,  at  43  dollars  apiece  ? 

26.  What  will  a  drove  of  150  sheep  come  to,  at  13 
shillings  per  head  ? 

27.  What  cost  105  acres  of  land,  at  15  dollars  per  acre? 

28.  How  much  will  135  yards  of  cloth  come  to,  at  18 
shillings  per  yard  ? 

29.  In  1  pound  there  are  16  ounces :  how  many  ounces 
are  there  in  246  pounds  ? 

30.  A  drover  sold  283  oxen,  at  38  dollars  per  head: 
how  much  did  he  receive  for  them  ? 

31.  If  you  walk  22  miles  per  day,  how  far  will  you 
walk  in  305  days  ? 

32.  In  one  day  there  are  24  hours:  how  many  hours 
are  there  in  365  clays  ? 


ART.  44.]  MULTIPLICATION.  45 

33.  Multiply  2345  by  175.  34.  Multiply  6207  by  235. 
35.  Mult.  10645  by  1262.  36.  Mult.  25271  by  2579. 
37.  Mult.  162537  by  21268.  38.  Mult.  425231  by  30765. 

27.*  What  will  15  caps  cost  at  18  shillings  a  piece? 

Analysis. — The    multiplier   15   is  a      Operation. 
composite  number,  the  factors  of  which  18 

are  5  and  3.     That  is,  1 5  =  5  x  3.     We  5 

first  multiply  the  multiplicand  by  the  90 

factor  5,  and  this  product  by  3.     The  _3 

last  product  is  the  answer.     Hence,  270  shill. 

44.  To  multiply  by  a  composite  number. 

Multiply  the  multiplicand  by  one  of  the  factors  of  the 
multiplier,  and  this  product  by  another,  and  so  on  till 
you  have  multiplied  by  all  the  factors.  The  last  product 
will  be  the  answer. 

OBS.  1. — A  composite  number  is  one  which  is  produced  by  multi- 
plying two  or  more  factors  together.  Thus,  14  =7x2;  55  =  11x5. 

2.  Some  numbers  may  be  resolved  into  more  than  two  factors; 
and  also  into  different  sets  of  factors.  Thus,  8  =  2x2x2=4x2. 

28.  What  are  the  factors  of  4,  6,  9,  10,  21,  35,  77  ? 

29.  Name  the  different  sets  of  factors  of  12,  16,  18, 

20,  27,  32,  63. 

30.  Name  the  different  sets  of  factors  of  24,  30,  36,  48, 

60,  72,  100. 

31.  Mult.  248by35usingthefactorsof  35.     Ans.  8680. 

32.  Multiply  l73by  28.  Ans.  4844. 

33.  Multiply  504  by  63.  Am.  31752. 

34.  Multiply  721  by  45.  Ans.  32445. 

35.  Multiply  1048  by  56.  Ans.  58688. 

36.  Multiply  2347  by  72.  Ans.  188984. 

37.  Multiply  4630  by  96.  Ans.  444480. 

38.  Multiply  25205  by  77.  Ans.  1940785. 

39.  Multiply  36042  by  108.  Ans.  3892536. 

QUEST. — 44.  How  do  you  multiply  by  ft  composite  number?  Ol>s.  Wba£ 
is  a  composite  number?  45.  How  multiply  by  10,  100, 1000,  &c. 


4S  MULTIPLICATION.  [SECT.  IV. 

45.  To  multiply  by  10,  100,  1000,  <fec. 

Annex  as  many  ciphers  to  the  multiplicand  as  there  are 
ciphers  in  the  multiplier,  and  the  number  thus  formed  will 
be  the  product  required. 

40.  What  will  10  dresses  cost,  at  18  dollars  apiece  ? 

Ans.   180  dolls. 

41.  26x100.  46.   469x10000. 

42.  37x100.  47.  523x100000. 

43.  51x1000.  48.  681x1000000. 

44.  226x1000.  49.   85612x10000. 

45.  341  x  1000.  50.   960305  x  100000. 
51.  What  will  20  wagons  cost,  at  67  dollars  apiece? 

Suggestion.  —  Since  multiplying          ~          . 
,        .   f  ,,  ,    .    *  Operation. 

by  ciphers  produces  ciphers,  we  omit  RH 

multiplying   by  the  0,  and   placing  2o 

the  significant   figure   2   under  the     ^  1340  dollar* 
right  hand   figure  of  the   multipli- 
cand, multiply  by  it  in  the  usual  way,  and  annex  a  cipher 
to  the  product.     The  answer  is  1340  dollars.     Hence, 

46.  When  there  are  ciphers  on  the  right  hand  of  the 
multiplier. 

Multiply  the  multiplicand  by  the  significant  figures  of 
the  multiplier,  and  to  this  product  annex  as  many  ciphers, 
as  are  found  on  the  right  hand  of  the  multiplier. 


(52.) 

85 

(53.) 
97 

(54.) 
123 

(55.) 
234 

200 

3000 

40000 

50000 

(56.) 
261 

(57.) 
329 

(58.) 
462 

(59.) 
571 

130 

2400 

35000 

460000 

QUEST. — 46.  When  there  are  ciphers  on  the  right  of  tho  multiplicand,  how 
do  von  iirnftApd ? 


ARTS.  45, — 47.]       MULTIPLICATION.  47 

60.  In  one  hour  there  are  60  minutes :  how  many  min- 
utes are  there  in  125  hours? 

61.  What  will  300  barrels  of  flour  cost  at  8  dollars  per 
barrel  ? 

62.  What  cost  400  yds.  of  cloth,  at  17  sbills.  per  yd.  ? 

63.  If  the  expenses  of  1  man  are  135  dollars  per  month, 
how  much  will  be  the  expenses  of  200  men  ? 

64.  If  1500  men  can  build  a  fort  in  235  days,  how  long 
will  it  take  one  man  to  build  it  ? 

47.  When  there  are  ciphers  on  the  right  of  the  mul- 
tiplicand. 

Multiply  the  significant  figures  of  the  multiplicand  by 
the  multiplier,  and  to  the  product  annex  as  many  ciphers, 
as  are  found  on  the  right  of  the  multiplicand. 

65.  What  will  43  building  lots  cost,  at  3500  dollars  a 
lot? 

Having  placed  the  multiplier  under          Operation. 
the  significant  figures  of  the  multipli-  3500 

cand,  multiply  by  it  as  usual,  and  to  43 

the  product  thus  produced,  annex  two  105 

ciphers,  because  there  are  two  ciphers  140 

on  the  right  of  the  multiplicand.  Ans.  150500  dolls. 

(66.)  (67.)  (68.)  (69.) 

1300  2400  21000  25000 

15  17  24  32 

70.  What  is  the  product  of  132000  multiplied  by  25  ? 

71.  What  is  the  product  of  430000  multiplied  by  34  ? 

72.  What  is  the  product  of  1520000  multiplied  by  43  ? 

73.  What  is  the  product  of  2010000  multiplied  by  52  ? 

74.  What  is  the  product  of  3004000  multiplied  by  61  ? 

QUEST. — 47.  When  there  are  ciphers  on  the  right  of  the  multiplicand,  how 
do  you  proceed  ? 


MULTIPLICATION. 


[SECT.  IV. 


48.  When  the  multiplier  and  multiplicand  both  have 
ciphers  on  the  right 

Multiply  the  significant  figures  of  the  multiplicand  by 
the  significant  figures  of  the  multiplier,  and  to  this  pro- 
duct annex  as  many  ciphers,  as  are  found  on  the  right  of 
both  factors. 

75.  Multiply  16000  by  3200, 

Having  placed  the  significant  figures 
of  the  multiplier  under  those  of  the  mul- 
tiplicand, we  multiply  by  them  as  usual, 
and  to  the  product  thus  obtained,  annex 
five  ciphers,  because  there  are  five  ci- 
paers  on  the  right  of  both  factors. 

Solve  the  following  examples : 
76.  2100X200. 
78.   12000X210. 
80.  38000X19000. 
82.  2800000X26000. 
84.   1000  miles  XI 40. 
86.  120  dollars  X  4200, 
88.  867  pounds X  424, 
90.  6726  rodsX627. 
92.  25268  pence X  4005, 


Operation. 
16000 
3200 
32 
48 
Ans.  51200000 


77.  3400X130. 
79.  25000X2600. 
81.  500000X42000. 
83.  140  yards  XI 6000. 
85.  20  dollars  X  35000. 
87.  75000  dolls.  X  365. 
89.  6830  feet  X  562. 
91.  7207  galls. X  807. 
93.  36074  tons X  4060. 
95.  703268X5346. 
97.  864325X6728. 
99.  4567832X27324. 
101.  7563057X62043. 


94.  376245X3164. 

96.   600400X7034. 

98.  432467X30005. 

100.  680539X80406. 

102.  Multiply    seventy-three    thousand   and    seven    by 
twenty  thousand  and  seven  hundred. 

103.  Multiply  six  hundred  thousand,  two  hundred  and 
three  by  seventy  thousand  and  seventeen. 

Qi'KST. — 48.  When  there  are  ciphers  on  the  right  of  both  the  multiplier  and 
multiplicand,  how  proceed  ? 


ASTS.  48— 50.]  m»MON.  49 

SECTION  V. 

• 

DIVISION. 

ART.  49.  Ex.  1.  How  many  lead  pencils,  at  2  cents 
apiece,  can  I  buy  for  10  cents? 

Solution. — :Since  2  cent 3  will  buy  1  pencil,  10  cents 
will  buy  as  many  pencils,  as  2  cents  are  contained  times  in 
10  cents  ;  and  2  cents  are  contained  in  10  cents,  5  times. 
I  can  therefore  buy  5  pencils, 

2.  A  father  bought  12  pears,  which  he  divided  equally 
among  his  3  children  :  how  many  pears  did  each  re- 
ceive ? 

Solution. — Reasoning  in  a  similar  manner  as  above,  it 
is  plain  that  each  child  will  receive  1  pear,  as  often  as  3 
is  contained  in  12  ;  that  is,  each  must  receive  as  many 
pears,  as  3  is  contained  times  in  12.  Now  3  is  contained  in 
12,  4  times.  Each  child  therefore  received  4  pears. 

OBS.  The  object  of  the  first  example  is  to  find  how  many  times 
one  given  number  is  contained  in  another.  The  object  of  the  second 
is  to  divide  a  given  number  into  several  equal  parts,  and  to  ascertain 
the  value  of  these  parts.  The  operation  by  which  they  are  solved 
is  precisely  the  same,  and  is  called  Division.  Hence, 

5O.  DIVISION  is  the  process  of  finding  how  many  times 
ore  given  number  is  contained  in  another. 

The  number  to  be  divided,  is  called  the  dividend. 

The  number  by  which  we  divide,  is  called  the  divisor. 

The  answer,  or  number  obtained  by  division,  is  called 
the  quotient,  and  shows  how  many  times  the  divisor  is 
contained  in  the  dividend. 


QUEST.— 50.  What  is  division  ?  What  is  the  number  to  be  divided,  called  T 
The  number  by  which  we  divide?  What  Is  the  answer  ca'led?  Whnt  does 
the  quotient  show  ? 


50  DIVISION.  [SECT.  V. 


Note.  —  The  term  quotient  is  derived  from  the  Latin  word 
which  signifies  how  often,  or  how  many  times. 

5  1  .  The  number  which  is  sometimes  left  after  division, 
is  called  the  remainder.  Thus,  when  we  say  4  is  con- 
tained in  21,  5  times  and  1  over,  4  is  the  divisor,  21  the 
dividend,  5  the  quotient,  and  1  the  remainder. 

OBS.  1.  The  remainder  is  always  less  than  the  divisor;  for  if  it 
were  equal  to,  or  greater  than  the  divisor,  the  divisor  could  be  con- 
tained once  more  in  the  dividend. 

2.  The  remainder  is  also  of  the  same  denomination  as  the  divi- 
dend ;  for  it  is  a  part  of  it. 

52.  Sign  of  Division  (-r).  The  sign  of  Division  is 
a  horizontal  line  between  two  dots  (—),  and  shows  that 
the  number  before  it,  is  to  be  divided  by  the  number 
after  it.  Thus,  the  expression  24—6,  signifies  that  24  is 
to  be  divided  by  6. 

Division  is  also  expressed  by  placing  the  divisor  under 
the  dividend  with  a  short  line  between  them.  Thus  the 
expression  •^  shows  that  35  is  to  be  divided  by  7,  and  is 
equivalent  to  35-T-7. 

53»  It  will  be  perceived  that  division  is  similar  in  prin- 
"  ciple  to  subtraction,  and  may  be  performed  by  it.  For 
instance,  to  find  how  many  times  3  is  contained  in  12, 
subtract  3  (the  divisor)  continually  from  12  (the  dividend) 
until  the  latter  is  exhausted  ;  then  counting  these  repeated 
subtractions,  we  shall  have  the  true  quotient.  Thus,  3 
from  12  leaves  9  ;  3  from  9  leaves  6  ;  3  from  6  leaves  3  ; 
3  from  3  leaves  0.  Now,  by  counting,  we  find  that  3  has 


QUKST.— 51.  What  is  the  number  called  which  is  sometimes  left  after  divi- 
sion? When  we  say  4  is  in  21,  5  times  and  1  over,  what  is  the  4  called  ?  The 
2 1  ?  The  5  ?  The  1  ?  Obs.  Is  the  remainder  greater  or  less  than  the  divisor  ? 
Why?  Of  what  denomination  is  it?  Why?  52.  What  is  the  sign  of  divi- 
Bion  ?  What  does  it  show  ?  In  what  other  way  is  division  expressed  ? 


ARTS.  51 — 53.]  DIVISION.  51 

been  taken  from  12,  4  times;  consequently  3  is  contained 
in  12,  4  times.     Hence, 

Division  is  sometimes  defined,  to  be  a  short  way  of  per- 
forming repeated  subtractions  of  the  same  number. 

OBS.  1.  It  will  also  be  observed  that  division  is  the  reverse  of 
multiplication.  Multiplication  is  the  repeated  addition  of  the  same 
number;  division  is  the  repeated  subtraction  of  the  same  number. 
The  product  of  the  one  answers  to  the  dividend  of  the  other :  but 
the  latter  is  always  given,  while  the  former  is  required. 

2.  When  the  dividend  denotes  things  of  one  kind,  or  denomina/ 
tion  only,  the  operation  is  called  Simple  Division. 

DIVISION   TABLE. 


1   is  in 

2   is  in 

3  is  in 

4   is  in 

5  is  in 

1,  once. 

2,  once. 

3,  once. 

4,  once. 

5,  once. 

2,          2 

4,           2 

6,           2 

8,           2 

10,          2 

3,           3 

6,           3 

9,           3 

12,           3 

15,           3 

4,           4 

8,          4 

12,           4 

16,           4 

20,           4 

5,           5 

10,           5 

15,           5 

20,           5 

25,           5 

6,           6 

12,           6 

18,           6 

24,           6 

30,           6 

1,          1 

14,           7 

21,           7 

28,          7 

35,           7 

8,           8 

16,           8 

24,           8 

32,           8 

40,           8 

9,           9 

18,           9 

27,           9 

36,           9 

45,           9 

10,         10 

20,         10 

30,         10 

40,         10   50,         10 

6   is   in 

7  is  in 

8   is   in 

9   is   in 

10   is   in 

6,  once. 

7,  once. 

8,  once. 

9,  once. 

10,  once. 

12,           2 

14,           2 

16,           2 

18,           2 

20,        2 

18,           3 

21,           3 

24,           3 

27,          3 

30,        3 

24,           4 

28,           4 

32,           4 

36,           4 

40,        4 

30,           5 

35,           5 

40,           5 

45,           5 

50,         5 

36,           6 

42,           6 

48,           6 

54,           6 

60,         6 

42,           1 

49,           7 

56,           7 

63,           7 

70,        7 

48,           8 

56,           8 

64,           8 

72,           8 

80,         8 

54,           9 

63,           9 

72,           9 

81,           9 

90,         9 

60,         10 

70,         10 

80,         10 

90,         10 

100,       10 

QUEST. — Obs.  When  the  dividend  denotes  things  of  one  denomination  onJy 
what  is  the  operation  called  ? 


52  DIVISION.  [SECT.  V 

SHORT   DIVISION. 

ART.  54.  Ex.  1.  How  many  yards  of  cloth,  at  2  dol- 
lars per  yard,  can  I  buy  for  246  dollars? 

Analysis. — Since  2  dollars  will  buy  1  yard,  246  dol- 
lars will  buy  as  many  yards,  as  2  dollars  are  contained 
times  in  246  dollars. 

Directions. — Write  the  divisor  on          Operation. 
the  left  of  the  dividend  with  a  curve      Divis01^  Divideild- 
line  between  them ;  then,  beginning 
at  the  left  hand,  proceed  thus :  2  is       ^Uot'  l 
contained  in  2,  once.     As  the  2  in  the  dividend  denotes 
hundreds,  the  1  must  be  a  hundred ;  we  therefore  write 
it  in  hundreds'  place  under  the  figure  divided.     2  is  con- 
tained in  4,  2  times ;  and  since  the  4  denotes  tens,  the  2 
must  also  be  tens,  and  must  be  written  in  tens'  place.    2  is 
n  6,  3  times.     The  6  is  units ;  hence  the  3  must  be  units, 
^nd  we  write  it  in  units'  place.     The  answer  is  123  yards. 

Solve  the  following  examples  in  a  similar  manner: 

2.  Divide  42  by  2.  6.  Divide  684  by  2. 

3.  Divide  69  by  3.  7.  Divide  4488  by  4. 

4.  Divide  488  by  4.  8.  Divide  3963  by  3. 

5.  Divide  555  by  5.  9.  Divide  6666  by  6. 

55.  When  the  divisor  is  not  contained  in  the  first 
igure  of  the  dividend,  we  must  find  how  many  times  it 
is  contained  in  the  first  two  figures. 

10.  At  2  dollars  a  bushel,  how  much  wheat  can  be 
bought  for  124  dollars? 

Since  the  divisor  2,  is  not  contained  in       Operation. 
the  first  figure  of  the  dividend,  we   find         2)124 
how  many  times  it  is  contained  in  the  first     Ans.  62  bu. 
two  figures.    Thus  2  is  in  12,  6  times;  set 
the  6  under  the  2.     Next,  2  is  in  4,  2  times.     The  an- 
swer is  62  bushels. 


ARTS.  54 — 57.  j  DIVISION.  53 

11.  Divide  142  by  2.  13.  Divide  1648  by  4. 

12.  Divide  129  by  3.  14.  Divide  2877  by  7. 

56»  After  dividing  any  figure  of  the  dividend,  if  there 
is  a  remainder,  prefix  it  mentally  to  the  next  figure  of  the 
dividend,  and  then  divide  this  number  as  before. 
Note. — To  prefix  means  to  place  before,  or  at  the  left  hand. 

15.  A  man   bought    42    peaches,   which   he   divided 
equally  among  his  3  children:  how  many  did  he  give  to 
each? 

When  we  divide  4  by  3,  there  is  1  re-        Operation. 
mainder.     This  we  prefix  mentally  to  the         3)42 
next  figure  of  the  dividend.     We  then  say,  14  Ans. 

3  is  in  12,  4  times. 

16.  Divide  56  by  4.  18.  Divide  456  by  6. 

17.  Divide  125  by  5.  19.  Divide  3648  by  8. 

57*  Having  obtained  the  first  quotient  figure,  if  the 
divisor  is  not  contained  in  any  figure  of  the  dividend,  place 
a  cipher  in  the  quotient,  and  prefix  this  figure  to  the  next 
one  of  the  dividend,  as  if  it  were  a  remainder. 

20.  If  hats  are  2  dollars   apiece,  how  many  can  be 
bought  for  216  dollars? 

As  the  divisor  is  not  contained  in  1,         Operation. 
the  second  figure  of  the  dividend,  we  2)216 

put  a  0  in  the  quotient,  and  prefix  the     Ans.  108  hats. 

1  to  the  6  as  directed  above.     Now  2 
13  in  16,  8  times. 

21.  Divide  2545  by  5.  23.  Divide  6402  by  6. 

22.  Divide  3604  by  4.  24.  Divide  4024  by  8. 
25.   A  man  divided  17  loaves  of  bread  equally  between 

2  poor  persons :  how  many  did  he  give  to  each  ? 

Suggestion. — Reasoning  as  before,  he  gave  each  as 
many  loaves  as  2  is  contained  times  in  17. 


54  DIVISION.  [SECT.  V. 

Thus,  2  is  contained  in  17,  8          Operation. 
times  and  1  over;  that  is,  after  2)17 

giving  them  8  loaves  apiece,  there      Quot.  8-1  remainder, 
is  one  loaf  left  which  is  not  divid-          Ans.  8-£  loaves. 
ed.     Now  2  is  not  contained  in  1 ; 

hence  the  division  must  be  represented  by  writing  the  2 
under  the  1,  thus  •£,  (Art.  52,)  which  must  be  annexed  to 
the  8.  The  true  quotient,  is  8£.  He  therefore  gave  ei<)kt 
and  a  half  loaves  to  each.  Hence, 

58»  When  there  is  a  remainder  after  dividing  the  last 
figure  of  the  dividend,  it  should  always  be  written  over  the 
divisor  and  annexed  to  the  quotient. 

Note. — To  annex  means  to  place  after,  or  at  the  right  hand. 

59»    When  the  process  of  dividing  is  carried  on  in  the 

mind,  and  the  quotient  only  is  set  down,  the  operation  is 
called  SHORT  DIVISION. 

6O»  From  the  preceding  illustrations  and  principles,  VTQ 
derive  the  following 

RULE   FOR   SHORT   DIVISION. 

I.  Write  the  divisor  on  the  left  of  the  dividend,  with  a 
curve  line  between  them. 

Beginning  at  the  left  hand,  divide  each  figure  of  the 
dividend  by  the  divisor,  and  place  each  quotient  figure 
under  the  figure  divided. 

II.  When  there  is  a  remainder  after  dividing  any  fig- 
ure, prefix  it  to  the  next  figure  of  the  dividend  and  divide 
this  number  as  before.     If  the  divizor  is  not  contained  in 

QUKST. — 59.  What  is  Short  Division  ?  60.  How  do  you  write  numbers 
for  short  division  1  Where  begin  to  divide  ?  Where  place  each  quotient  fig- 
ure? When  there  is  a  remainder  after  dividing  a  figure  of  the  dividend, 
what  must  be  done  with  it  ?  If  tho  divisor  is  not  contained  in  a,  figure  of  the 
dividend,  how  proceed  7  When  there  is  a  remainder,  after  dividing  the  lust 
figure  of  the  dividend,  what  must  be  dune  with  it  ? 


ARTS.  58 — 61.]  DIVISION.  55 

any  figure  of  the  dividend,  place  a  cipher  in  the  quotient, 
and  prefix  this  figure  to  the  next  one  of  the  dividend,  as  if 
it  were  a  remainder.  (Arts.  56,  57.) 

Ill,  When  there  is  a  remainder  after  dividing  the  last 
fyure,  write  it  over  the  divisor  and  annex  it  to  the  quotient. 

Ol»  PROOF. — Multiply  the  divisor  ly  the  quotient,  to 
the  product  add  the  remainder,  and  if  the  sum  is  equal  to 
the  dividend,  the  work  is  right. 

OBS.  Division  may  also  be  proved  by  subtracting  the  remainder, 
if  any,  from  the  dividend,  then  dividing  the  result  by  the  quotient. 

EXAMPLES    FOR    PRACTICE. 

1.  Divide  426  by  3.  10.  Divide  3640  by  5. 

2.  Divide  506  by  5.  11.  Divide  6210  by  4. 

3.  Divide  304  by  4.  12.  Divide  7031  by  7. 

4.  Divide  450  by  6.  13.  Divide  2403  by  6. 

5.  Divide  720  by  7.  14.  Divide  8131  by  9. 

6.  Divide  510  by  9.  15.  Divide  7384  by  8. 

7.  Divide  604  by  5.  16.  Divide  8560  by  7. 

8.  Divide  760  by  8.  17.  Divide  7000  by  8. 

9.  Divide  813  by  7.  18.  Divide  9100  by  9. 

19.  How  many  pair  of  shoes,  at  2  dollars  a  pair,  can 
you  buy  for  126  dollars  ? 

20.  How  many  hats,  at  4  dollars  apiece,  can  be  bought 
for  168  dollars? 

21.  A  oan  bought  144  marbles  which  he  divided  equally 
among  his  6  children :  how  many  did  each  receive  ? 

22.  A  man  distributed  360  cents  to  a  company  of  poor 
children,  giving  8  cents  to  each  :  how  many  children  were 
there  in  the  company  ? 

23.  How  many  yards  of  silk,  at  6  shillings  per  yard, 
can  I  buy  for  450  shillings  ? 

UUEST.— 61.  How  is  division  proved?  Obs.  What  other  way  of  proving 
^i\  ition  is  mentioned? 


o6  DIVISION.  [SECT.  V. 

24.  A  man  having  600  dollars,  wished  to  lay  it  out 
in  flour,  at  7  dollars  a  barrel :  how  man)-  *vhole  barrels 
could  he  buy,  and  how  many  dollars  would  he  have  left  ? 

25.  If  you  read  9  pages   each  day,  how  long  will  it 
tike  you  to  read  a  book  through  which  has  82  R  pages  ? 

26.  If  I  pay  8  dollars  a  yard  for  broadcloth,  ho^r  Hsaiiy 
yards  can  I  buy  for  1265  dollars? 

27.  If  a  stage  coach  goes  at  the  rate  of  8  milfs  pfti 
hour,  how  long  will  it  be  in  going  1560  miles  ? 

28.  If  a  ship  sails  9  miles  an  hour,  how  long  will  k 
be  in  sailing  to  Liverpool,  a  distance  of  3000  miles  ? 

LONG  DIVISION. 

ART.  62.  Ex.  1.  A  man  having  156  dollars  laid  it 

out  in  sheep  at  2  dollars  apiece :  how  many  did  he  buy  ! 

Analysis. — Reasoning  as  before,  since  2  dollars  will 

buy  1  sheep,  156   dollars  will  buy  as  many  sheep  as  2 

dollars  are  contained  times  in  156  dollars. 

Directions. — Having  written  the  di-          Operation. 
visor  on  the  left  of  the  dividend  as  in    Divis-  Dijid-  <*"<*• 
short  division,  proceed  in  the  follow-  14  • 

ing  manner :  — — 

First.  Find   how  many  times   the  ^ 

divisor  (2)  is  contained  in  (15)  the 
first  two  figures  of  the  dividend,  and  place  the  quotient 
figure  (7)  on  the  right  of  the  dividend  with  a  curve  line 
between  them.  Second.  Multiply  the  divisor  by  the 
quotient  figure,  (2  times  7  are  14,)  and  write  the  product 
(14)  under  the  figures  divided.  Third.  Subtract  the 
product  from  the  figures  divided.  (The  remainder  is  1.) 
Fourth.  Bringing  down  the  next  figure  of  the  dividend, 
and  placing  it  on  the  right  of  the  remainder  we  have  16. 
Now  2  is  contained  in  16,  8  times;  place  the  8  on  the 
right  hand  of  the  lust  quotient  figure,  and  multiplying 


ARTS.  62, 63.]  DIVISION.  57 

the  divisor  by  it,  (8  times  2  are  16,)  set  the  product  undei 
the  figures  divided,  and  subtract  as  before.  Therefore  156 
dollars  will  buy  78  sheep,  at  2  dollars  apiece. 

63.  When  the  result  of  each  step  in  the  operation  u 
wt  down,  the  process  of  dividing  is  called  LONG  DIVISION. 

It  is  the  same  in  principle  as  Short  Division.  The 
only  difference  between  them  is,  that  in  Long  Division 
the  result  of  each  step  in  the  operation  is  written  down, 
while  in  Short  Division  we  carry  on  the  whole  process 
in  the  mind,  simply  writing  down  the  quotient. 

Note. — To  prevent  mistakes,  it  is  advisable  to  put  a  dot  under 
each  figure  of  the  dividend,  when  it  is  brought  down. 

Solve  the  following  examples  by  Long  Division: 

2.  Divide  195  by  3.  Ans.  65. 

3.  Divide  256  by  2.  6.  Divide  2665  by  5. 

4.  Divide  1456  by  4.  7.  Divide  4392  by  6. 

5.  Divide  5477  by  3.  8.  Divide  6517  by  7. 

OBS.  When  the  divisor  is  not  contained  in  the  first  two  figures  of 
the  dividend,  find  how  many  times  it  is  contained  in  the  first  three, 
or  the/bices/  figures  which  will  contain  it,  and  proceed  as  before. 

9.  How  many  times  is  13  contained  in  10519? 

Tims,  13  is  contained  in  105,          Operation. 
8  times;  set  the  8  in  the  quo-     13)10519(809W  Ann. 
tient  then  multiplying  and  sub-  104 

tracting,    the    remainder    is    1.  119 

Bringing  down  the  next  figure  117 

we  have  11  to  be  divided  by  13.  2  rem. 

But  I  3  is  not  contained  in  1 1  ; 

therefore  we  put  a  cipher  in  the  quotient,  and  I  ring  down 
the  next  figure.  (Art.  57.)  Then  13  is  contained  in  119, 

Qi  KST.— 03.  What  is  long  division?  What  is  the  difference  between  lonn 
and  short  division  ? 


58  DIVISION.  [SECT.  V. 

9  limes.  Set  the  9  in  the  quotient,  multiply  and  sub- 
tract, and  the  remainder  is  2.  Write  the  2  over  the  di- 
visor, and  annex  it  to  the  quotient.  (Art.  58.) 

64.  After  the  first   quotient  figure  is  obtained,  for 
fuck  figure  of  the  dividend  which  is  brought  down,  either 
a  significant  figure  or  a  cipher  must  be  put  in  the  quotient. 

Solve  the  following  examples  in  a  similar  manner : 

10.  Divide  15425  by  11.  Ans.  1402-fr. 

11.  Divide  31237  by  15.  Ans.  2082-^. 

65.  From  the  preceding  illustrations  and  principles, 
we  derive  the  following 

RULE  FOR  LONG  DIVISION. 

I.  Beginning  on  the  left  of  the  dividend,  find  how  many 
times  the  divisor  is  contained  in  the  fewest  figures  that  will 
contain  it,  and  place  the  quotient  figure  on  the  right  of 
the  dividend  with  a  curve  line  between  them. 

II.  Multiply  the  divisor  by  this  figure  and  subtract 
the  product  from  the  figures  divided ;   to  the  right  of 
the  remainder  bring  down  the  next  figure  of  the  dividend, 
and  divide  this  number  as  before.     Proceed  in  this  man 
ner  till  all  the  figures  of  the  dividend  are  divided. 

III.  When  there  is  a  remainder  after  dividing  the  last 
figure,  write  it  over  the  divisor,  and  annex  it  to  the  quo- 
tient, as  in  short  division. 

OBS.  1.  Long  Division  is  proved  in  the  same  manner  as  Short 
Division. 

2.  When  the  divisor  contains  but  one  figure,  the  operation  by 
Short  Division  is  the  most  expeditious,  and  should  therefore  be 
practiced ;  but  when  the  divisor  contains  two  or  more  figures,  it  will 
generally  be  the  most  convenient  to  divide  by  Long  Division. 

QUEST. — 65.  How  do  you  divide  in  long  division?  Whore  place  the  quo 
icnt  ?  After  obtaining  the  first  quotient  figure,  how  proceed  ?  When  there  is 
i  remainder  after  dividing  the  last  figure  of  the  dividend  what  must  be  done 
with  it?  Obs.  How  is  long  division  proved?  When  should  short  division 
5)t-  used  ?  When  long  division* 


ARTS.  64,  65.]  DIVISION.  59 

EXAMPLES  FOR  PRACTICE. 

1.  Divide  369  by  8.  10.  Divide  675  by  25. 

2.  Divide  435  by  9.  11.  Divide  7'42  by  31. 

3.  Divide  564  by  7.  12.  Divide  798  by  37. 

4.  Divide  403  by  10.  13.  Divide  334  by  42. 

5.  Divide  641  by  11.  14.  Divide  960  by  48. 

6.  Divide  576  by  12.  15.  Divide  1142  by  53. 

7.  Divide  274  by  13.  16.  Divide  2187  by  67. 

8.  Divide  449  by  14.  17.  Divide  3400  by  75. 

9.  Divide  617  by  15.  18.  Divide  4826  by  84. 

19.  How  many  caps,  at  7  shillings  apiece,  can  I  buy 
for  168  shillings? 

20.  How  many  pair  of  boots,  at  5  dollars  a  pair,  can 
be  bought  for  175  dollars  ? 

21.  A  man  laid  out  252  dollars  in  beef,  at  9  dollars  a 
barrel :  how  many  barrels  did  he  buy  ? 

22.  In  1 2  pence  there  is  1  shilling :  how  many  shillings 
are  there  in  198  pence? 

23.  In  20  shillings  there  is  1  pound  :  how  many  pounds 
are  there  in  215  shillings? 

24.  In  16  ounces  there  is  1  pound:  how  many  pounds 
are  there  in  268  ounces  ? 

25.  How  many  trunks,  at  15  shillings  apiece,  can  be 
bought  for  255  shillings  ? 

26.  If  27  pounds  of  flour  will  last  a  family  a  week, 
how  long  will  810  pounds  last  them? 

27. .How  many  yards  of  broadcloth,  at  23  shillings  per 
yard,  can  be  bought  for  756  shillings  ? 

28.  If  it  takes  18  yards  of  silk  to  make  a  dress,  how 
many  dresses  can  bd  made  from  1350  yards? 

29.  How  many  sheep,  at  19  shiLj.gs  per  head,  can  be 
bought  for  1539  shillings? 

30.  A  farmer  having   1840  dollars,  laid  it  out  in  land, 
at  25  dollars  per  acre :  how  many  acres  did  he  buy  ? 


60  DIVISION.  [SECT.  V. 

31.  A  man  wishes  to  invest  2562  dollars  in  Railroad 
stock  :  how  many  shares  can  he  buy,  at  42  dollars  per 
share  ? 

32.  In  1  year  there  are  52  weeks :  how  many  years 
are  there  in  1640  weeks  ? 

33.  In  one  hogshead  there  are  63  gallons :  how  many 
hogsheads  are  there  in  3065  gallons  ? 

34.  If  a  man  can  earn  75  dollars  in  a  month,  how  many 
months  will  it  take  him  to  earn  3280  dollars  ? 

35.  If  a  young  man's  expenses  are  83  dollars  a  month, 
how  long  will  4265  dollars  support  him  ? 

36.  A  man  bought  a  drove  of  95  horses  for  4750  dol- 
lars :  how  much  did  he  give  apiece  ? 

37.  If  a  man  should  spend  16  dollars  a  month,  how 
long  will  it  take  him  to  spend  172  dollars? 

38.  A  garrison  having  2790  pounds  of  meat,  wished  to 
have  it  last  them  31  days:  how  many  pounds  could  they 
eat  per  day  ? 

39.  How  many  times  is  54  contained  in  3241,  and  how 
many  over  ? 

40.  How  many  times  is  68  contained  in  7230,  and  how 
many  over  ? 

41.  How  many  times  is  39  contained  in  1042,  and  how 
many  over? 

42.  How  many  times  is  47  contained  in  2002,  and  how 
many  over  ? 

43.  What  is  the  quotient  of  1704  divided  by  56  ? 

44.  What  is  the  quotient  of  2040  divided  by  60  ? 

45.  What  is  the  quotient  of  2600  divided  by  49  ? 

46.  What  is  the  quotient  of  2847  divided  by  81  ? 

47.  Divide  1926  by  75.          51.  Divide  9423  by  105. 

48.  Divide  2230  by  85.          52,  Divide  13263  by  112. 
40.  Divide  6243  by  96.          53.  Divide  26850  by  123. 
50.  Divide  8461  by  99.          54.  Divide  48451  by  224. 


ARTS.  66,  67.]  DIVISION.  61 

46.*  How  many  balls  at  15  pence  apiece,  can  be 
bought  for  146  pence  2 

Analysis. — This  divisor  being  com-          Operation. 
posite,  whose  factors  are   3  and  5,  we         3)149 
first  divide  by  3,  and  this  quotient  by  5.          5)49"  g  rem 

To  find  the  true  remainder,  multiply  -— ' 

each  remainder  bv  all  the  divisors  pre-       . 

,....«  i>-..  M  <&**•  9  and  14  r. 

ceding  the  division  from  which  it  arose, 

and  to  the  sum  of  the  products  add  the  first  remainder; 
the  result  will  be  the  true  remainder.  In  this  example 
the  only  preceding  divisor  is  3  ;  now  the  last  remainder 
4  x  3  =  12,  and  12  +  2  =  14  the  true  remainder.  Hence, 

66.  To  divide  by  a  composite  number. 

Divide  the  dividend  by  one  of  the  factors,  then 
divide  the  quotient  thus  obtained  by  another  factor,  and 
so  on  till  all  the  factors  are  employed.  The  last  quotient 
will  be  the  answer. 

47.  Divide  231  by  21,  using  its  factors.          Ans.  11. 

48.  Divide  195  by  16,  using  its  factors.    Ans.  12  and  3  r. 

49.  Divide  256  by  24,  using  its  factors.     Ans.  10  and  16  r. 

50.  Divide  365  by  48,  using  its  factors.     Ans.  7  and  29  r. 

51.  Divide  410  by  45,  using  its  factors.     Ans.  9  and  5  r. 

52.  Divide  217  by  63,  using  its  factors.  Ans.  3  and  28  r. 

53.  Divide  561  by  56,  using  its  factors.     Ans.  10  and  1  r. 

54.  Divide  893  by  72,  using  its  factors.  Ans.  12  and  29  r. 

55.  Divide  1275  by  96,  using  its  factors.  Ans.  13  and  2  r. 

67.  To  divide  by  10,  100,  1000,  &c. 

Cut  off  as  many  figures  from  the  right  hand  of  tJte 
dividend  as  there  are  ciphers  in  the  divisor.  The  remain- 
ing figures  of  the  dividend  will  be  the  quotient,  and  those 
cut  off  the  remainder. 

56.  Divide  1325  by  10.        57.  Divide  4626  by  100. 
58.  Divide  5633  by  1000.      59.  Divide  8465  by  1000. 
60.  Divide  26244  by  1000.  61.  Divide  136056  by  10000. 


DIVISION. 


[SECT.  V. 


62.  Divide  2443667  by  100000. 

63.  Divide  23454631  by  1000000. 

68.  When  there  are  ciphers  on  the  right  hand  of  the 
divisor. 

Cut  off  the  ciphers  from  the  divisor  ;  also  cut  off  as 
many  figures  from  the  right  of  the  dividend.  Then  divide 
the  remaining  figures  of  the  dividend  by  the  remaining  fig- 
ures of  the  divisor,  and  the  result  will  be  the  quotient. 

Finally,  annex  the  figures  cut  off  from  the  dividend  to 
the  remainder,  and  the  number  thus  formed  ivill  be  the  true 
remainder. 

64.  At  200  dollars  apiece,  how  many  carriages  can  be 
bought  for  4765  dollars  ? 

Having  cut  off  the  two  ciphers  on 
the  right  of  the  divisor,  and  two  fig- 
ures on  the  right  of  the  dividend,  we 
divide  the  47  by  2  in  the  usual  way. 

65.  Divide  2658  by  20. 


66.  3642  by  30. 
68.  76235  by  1400. 
70.  93600  by  2000. 
72.   23148  by  1200. 
74.  50382  by  1800. 
76.  894000  by  2500. 
78.  7450000  by  420000. 


Operation. 
2|00)47|65 
Ans.  23 — 165  rem. 


80.  348676- 
82.  762005 
84.  607507- 


235. 
401. 
1623. 


86.  4367238-^2367. 
88.  8230732^3478. 
90.  93670858-r67213. 


Ans.  132  and  18  rem.,  or 

67.  6493  by  200. 
69.  82634  by  1600. 
71.  14245  by  3000. 
73.  42061  by  1500. 
75.  88317  by  2100. 
77.  9203010  by  3100. 
79.  9000000  by  300000. 
81.  467342  —  341. 
83.  506725  —  603. 
85.  736241  —  2764. 
87.  6203451-^3827. 
89.  8235762-^-42316. 
91.  98765421-7-84327. 


QUEST.— 68.  When  there  are  ciphers  on  the  right  of  th«  divisor,  how  pro 
cecd  ?    What  is  to  be  done  with  figures  cut  oft' from  the  dividend  7 


ART.  71.]  DEFINITIONS.  63 

ARITHMETICAL  TERMS. 
7  1  •  Numbers  are  divided  into  abstract  arid  concrete, 

1.  Abstract  numbers  are  numbers  used  without  appli- 
cation to  any  object ;  as  two,  three,  four,  five,  &e. 

2.  Concrete  numbers  are  numbers  applied  to  some  par- 
ticular object ;  as  two  peaches,  three  pounds,  &c. 

3.  Numbers  are  also  divided  into  prime  and  composite. 

4.  A  prime  number  is  one  which  cannot  be  produced 
by  multiplying  any  two  or  more  numbers  together;  or 
which  cannot  be  exactly  divided  by  any  whole  number, 
except  a  unit  and  itself.     Thus,  1,  2,  3,  5,  7,  11,  13,  17, 
19,  23,  29,  31,  37,  41,  43,  47,  &c.,  are  prime  numbers. 

OBS.  1.  The  least  divisor  of  every  number  is  a  prime  number. 

2.  One  number  is  said  to  be  prime  to  another,  when  a  unit  is  the 
only  number  by  which  both  can  be  divided  without  a  remainder. 

3.  The  number  of  prime  numbers  is  unlimited.     All  below  fifty 
are  given  above.     The  pupil  can  easily  point  out  others. 

5.  A  composite  number  is  one  which  is  produced  by  mul- 
tiplying two  or  more  factors  together.   Thus,  12=4  x  3. 

6.  An  even  number  is  one  which  can  be  divided  by  2 
without  a  remainder;  as,  4,  6,  8,  10. 

7.  An  odd  number  is  one  which  cannot  be  divided  by 
2  without  a  remainder;  as,  1,  3,  5,  7,  9,  15. 

8.  One  number  is  a  measure  of  another,  when   the 
former   will    divide   the    latter,    without    a   remainder. 
Thus  2  is  a  measure  of  4 ;  3  is  a  measure  of  6. 

9.  A  common  measure  is  a  number,  which  will  divide 
two  or  more  numb.ers,  without  a  remainder.     Thus,  2  is  a 
common  measure  of  4,  6,  and  8. 

10.  The  aliquot  parts  of  a  number  are  the  parts  by 
which  it  can  be  divided  without  a  remainder.     Thus,  3 
and  7  are  aliquot  parts  of  21. 


QUEST. —71.  What  are  abstract  numbers?    Concrete?    Prime?     Compo- 
site?   An  oven  number?    Au  odd  number?    A  common  measure? 


64  COMMON    DIVISOR.  [SECT.  V. 

GREATEST   COMMON   DIVISOR. 

72.  A  Common  Divisor  is  a  number  which  will  di- 
vide two  or  more  numbers  without  a  remainder.     Thus, 
2  is  a  common  divisor  of  4,  6,  8,  12,  10. 

73.  The  Greatest  'Common  Divisor  of  two  or  more 
numbers,  is  the  greatest  number  which  will  divide  each 
of  them  without  a  remainder.     Thus,  0  is  the  greatest 
common  divisor  of  12,  18,  and  24. 

1.  What  is  the  greatest  common  divisor  of  30  and  42  ? 
Suggestion. — Dividing  42  by  30,  the          ^          . 

remainder  is  12;  then  dividing  30  (the 
preceding  divisor)    by   12  (the  last  re-     30)42(1 
inainder)  the  remainder   is  6  ;  finally,  — 

dividing   12   (the  preceding  divisor)  by  ''    ^ 

6  (the  last  remainder)  nothing  remains;  ~ir  \12f2 

consequently  6,  the  last  divisor,  is  tho  ',^ 

greatest  common  divisor.     Hence, 

7  4.  To  find  the  greatest  common  divisor  of  two  num- 
bers. 

Divide  tlie  greater  number  by  the  less  /  then  divide 
the  preceding  divisor  by  the  last  remainder,  and  so  on, 
till  nothing  remains.  The  last  divisor  will  be  the  great- 
est common  divisor. 

2.  What  is  the  greatest  com.  divisor  of  56  and  140  ? 

3.  What  is  the  greatest  com.  divisor  of  116  and  203  ? 

4.  Find  the  greatest  common  divisor  of  148  and  185. 

5.  Find  the  greatest  common  divisor  of  237  and  395. 

C.  What  is  the  greatest  com.  divisor  of  122  and  427  ? 

& 

75.  To  find  the  greatest  common  divisor  of  more  than 
two  numbers? 

First  find  the  greatest  common  divisor  of  any  two  of 

QUEST. — What  are  aliquot  parts  of  a  number?  72.  A  common  divisor? 
73.  The  greatest  common  divisor  of  two  or  more  r  umbers?  74.  How  find 
tlie  greatest  common  divisor  of  two  numbers?  75.  Of  mon?  than  two? 


ARTS.  72 — 79.]       COMMON  MULTIPLE.  65 

the  given  numbers  •  then,  that  of  the  common  divisor 
thus  obtained  and  of  another  number,  and  so  on  through 
all  the  given  numbers.  The  last  common  divisor  found, 
will  be  the  one  required. 

7.  Find  the  greatest  com.  divisor  of  45,  63  and  108. 

8.  Find  the  greatest  com.  divisor  of  32,  48  and  200. 

9.  Find  the  greatest  com.  divisor  of  256,  372  and  522. 

LEAST   COMMON   MULTIPLE. 

76.  A  multiple  is  a  number  which  can  be  divided  by 
another  number  without  a  remainder.     Thus,  4  is  a  mul- 
tiple of  2  ;  10  is  a  multiple  of  5. 

77.  A  common  multiple  is  a  number  which  can  be 
divided  by  two  or  more  numbers  without  a  remainder. 
Thus,  12  is  a  common  multiple  of  2,  3,  and  4. 

78.  The  continued  product  of  two  or  more  numbers 
is  always  a  common  multiple  of  those  numbers. 

7  9.  The  least  common  multiple  of  two  or  more  num- 
bers, is  the  least  number  which  can  be  divided  by  each  of 
them  without  a  remainder.  Thus,  12  is  the  least  com- 
mon multiple  of  4  and  6. 

10.  Find  the  least  common  multiple  of  6,  10,  and  12. 
Suggestion. — Write  the  given  num-  Operation. 

bers  in  a  line,  and,  dividing  by  2  the  2>I6"10"19 

smallest  number  that  will  divide  any  —^ — rr — 

two  of  them  without  a  remainder,  set  '~ — - — 

the  quotients  3,  5,  and  6  in  a  line  be-  _ 
low.     Next  divide  this  line  by  3  and 

set  the  quotients  and  undivided  number  5  in  a  line  as 

before.     Finally,  multiply  all  the  divisors  into  the  quo- 


QUEST.— 76.  What  is  a  multiple?     77.  A  common  multiple  ?    79.  What  is 
!IH  lea*!,  common  multiulc  of  two  or  more  numbers? 


66  COMMON    MULTIPLE.  [SECT.  Y. 

tients  and  undivided  numbers  in  the  last  line,  and  the 
product  is  the  answer  required.     Hence, 

8O.  To  find  the  least  common  multiple  of  two  or 
more  numbers. 

I.  Write  the  given  numbers  in  a  horizontal   line,  and 
divide  by  the  smallest  number  which  will  divide  any  two 
cr  more  of  them  without  a  remainder,  setting  the  quotients 
ind  undivided  numbers  in  a  line  below. 

II.  Divide  this  line  and  set  down  the  results  as  before  ; 
thus  continue  the  operation  till  there  are  no  two  numbers 
which  can  be  exactly  divided    by   any   number  greater 
than  1. 

III.  Finally,  multiply  all  the  divisors  and  numbers  in 
the  last  line  together,  and  the  product  will  be  the  least 
common  multiple. 

11.  Find  the  least  common  multiple  of  16  and  20. 

12.  Find  the  least  common  multiple  of  14,  21,  and  28. 

13.  Find  the  least  common  multiple  of  18,  5,  12,  10. 

14.  Find  the  least  common  multiple  of  16,  15,  8,  36. 

15.  Find  the  least  common  multiple  of  12,  18,  6,  8. 

16.  Find  the  least  common  multiple  of  20,  32,  50,  35. 

17.  Find  the  least  common  multiple  of  45,  18,  56,  64. 

18.  Find  the  least  common  multiple  of  15,  17,  25,  51. 

19.  Find  the  least  common  multiple  of  10,  33,  18,  90. 

20.  Find  the  least  common  multiple  of  11,  22,  13,  39. 

21.  Find  the  least  common  multiple  of  25,  36,  45,  96. 

22.  Find  the  least  common  multiple  of  108,  256,  320. 

23.  Find  the  least  common  multiple  of  8, 12, 16,  36,  44. 

24.  Find  the  least  common  multiple  of  288,  360, 1728. 

25.  Find  the  least  common  multiple  of  136,  458,  890. 

26.  Find  the  least  common  multiple  of  256,  576,  2000. 

27.  Find  the  least  common  multiple  of  820,  575,  3500. 

QUEST. — 80.  How  is  the  least  common  multiple  of  two  or  more  numbers 
found  ? 


ARTS.  80 — 84.]  .  FRACTIONS.  67 


SECTION    VI. 

FRACTIONS. 

81*  When  a  number  or  thing  is  divided  into  equal 
parts,  the  parts  are  called  fractions.  If  divided  into  two 
equal  parts,  one  of  these  parts  is  called  one  half;  if  di- 
vided into  three  equal  parts,  one  of  these  parts  is  called 
one  third  ;  if  divided  into  four  equal  parts,  one  of  the 
parts  is  called  one  fourth,  or  one  quarter  ;  if  into  ten, 
tenths  ;  if  into  a  hundred,  hundredth*,  &c.  Hence, 

82.  A  FRACTION  denotes  apart  or  parts  of  a  number 
or  tiling.     An  Integer  is  a  whole  number. 

Note. — The  term  fraction,  is  derived  from  the  Latin  f radio,  which 
signifies  the  act  of  breaking,  a  broken  part  or  piece.  Hence,  fractions 
are  sometimes  called  broken  numbers. 

83.  Fractions  are  divided  into  two  classes,  Common 
and  Decimal.     (For  the  illustration  of  Decimal  Fractions 
see  Practical  Arithmetic.) 

8  4«  Common  fractions  are  those  which  arise  from  di- 
viding an  integer  into  any  number  of  equal  parts. 

They  are  expressed  by  two  numbers,  one  placed  over 
the  other,  with  a  line  between  them.  For  example,  one 
half  is  written  thus,  | ;  one -third,  1 ;  nine  tenths,  T\. 

The  number  below  the  line  is  called  the  denominator, 
and  shows  into  how  many  parts  the  number  is  divided. 

The  number  above  the  line  is  called  the  numerator, 
and  shows  how  many  parts  are  expressed  \>y  the  fraction. 
Thus,  in  the  fraction  f ,  the  denominator  3,  shows  that 
tKe  number  is  divided  into  three  equal  parts  ;  the  nume- 
rator 2,  shows  that  two  of  those  parts  are  expressed  by 
the  fraction. 


QUEST.— 81.  What  is  meant  by  one  half?    By  one  third?    82.  What  is  a 
fraction  ?    An  integer  ? 


68  FRACTIONS.  [SECT.  VL 

The  numerator  and  denominator  together  are  called 
the  terms'of  the  fraction. 

OBS.  The  number  below  the  line  is  called  the  denominator,  be- 
cause it  gives  the  name  or  denomination  to  the  fraction  j  as  halves, 
thirds,  fifths,  &c. 

The  number  above  the  line  is  called  the  numerator,  because  it 
numbers  the  parts,  or  shows  how  many  parts  are  expressed  by  the 
fraction. 

85.  Common  Fractions  are  divided  into  proper,  im- 
proper, simple,  compound,  complex,  and  mixed  numbers. 

A.  proper  fraction  is  a  fraction  whose  numerator  is  less 
than  its  denominator ;  as,  |,  f ,  £. 

An  improper  fraction  is  one,  whose  numerator  is  equal 
to,  or  greater  than  its  denominator  ;  as,  f ,  J. 

A  simple  fraction  is  a  fraction  which  has  but  one  nu- 
merator and  one  denominator,  and  may  be  proper  or  im- 
proper ;  as,  f,  f . 

A  compound  fraction  is  a  fraction  of  a  fraction  ;  as, 
I  of  t  off 

A  complex  fraction  is  one  which  has  a  fraction  in  its 

numerator  or  denominator,  or  in  both  ;  as,  -—,— -,  -~,~- 

5     oi  «f   - 

A  mixed  number  is  a  whole  number  and  a  fraction 
written  together;  as,  4|,  25}i. 

86.  The  value  of  a  fraction  is  the  quotient  of  .the  nu- 
merator divided  by  the  denominator.     Thus  the  value  of 
•|  is  two;  off  is  one  ;  of  ^  is  one  third;  &e.     Hence, 

If  the  numerator  and  denominator  are  equal,  the  value 
is  a  unit  or  one.  Thus,  J  —  1,  ^  =  1,  &c. 

If  the  numerator  is  greater  than  the  denominator,  the 
value  is  greater  than  one.  Thus,  f  =  2,  f  —  If. 

If  the  numerator  is  less  than  the  denominator,  the  value 
is  less  than  one.  Thus,  £  =  1  third  of  1,  £  —  ^fifths  of  1. 

QUEST.— 85.  How  are  common  fractions  divided  ?  What  is  a  proper  frac- 
tion ?  Improper  ?  "Simple  ?  Complex  ?  A  mixed  number  ? 


ART.  86 — 89.]  FRACTIONS.  69 

REDUCTION   OF   FRACTIONS. 

87.  Reduction  of  Fractions  is  the  process  of  chang- 
ing their  terms  into  others,  without  altering  the  value  of 
the  fractions. 

CASE  I. — Reducing  fractions  to  their  lowest  terms. 

88.  A  fraction  is    said  to  be  reduced  to  its  lowest 
terms,  when  its  numerator  and  denominator  are  expressed 
in  the  smallest  numbers  possible. 

Ex.  1.  Reduce  £  to  its  lowest  terms. 

Suggestion. — Dividing  both  _.     .   ~         .. 

^  • 5  First  Operation. 

terms  of  the  fraction  by  2,  it 

becomes  J.  Then,  dividing  2H  =  *:  and  2)*=*'  Ans' 
both  by  2  again,  we  obtain  •£,  whose  terms  are  the  low- 
est to  which  the  given  fraction  can  be  reduced. 

Or.  divide  both   terms  by   their      a        ,  -.        ,. 

J         .       Second  Operation. 
greatest  common  divisor,  which  is 

4,  and  the  given  fraction  will  be  re-  '8      *"" 

duced  to  its  lowest  terms  by  a  single  division.    Hence, 

89.  To  reduce  a  fraction  to  its  lowest  terms. 
Divide  the  numerator  and  denominator  by  any  number 

-which  will  divide  them  both  without  a  remainder ;  then 
divide  this  result  as  before,  and  so  on  until  no  number 
greater  than  1  will  exactly  divide  them ;  the  last  two 
quotients  will  be  the  lowest  terms  to  which  the  given  frac- 
tion can  be  reduced. 

Or,  divide  both  the  numerator  and  denominator  by 
their  greatest  common  divisor  ;  and  the  quotients  will  be 
the  lowest  terms  of  the  given  fraction, 

OBS.  The  value  of  a  fraction  is  not  altered  by  reducing  it  to  its 


OBTEST. — 87.  What  is  reduction  of  fractions?     89.  How  is  a  fraction  re- 
duced to  its  lowest  terms  ? 


70  FRACTIONS.  [SECT.  VI. 

lowest  terms ;  for  the  numerator  and  denominator  are  divided  by. 
the  same  number. 

2.  Reduce  T%-  to  its  lowest  terms.     Ans.  \. 

3.  Reduce  |f.  4.  Reduce  ¥V 
5.  Reduce  if.  6.  Reduce  ±\. 
7.  Reduce  f£.                                8.  Reduce  -|f. 
9.  Reduce  if|.                            10.  Reduce  -/fy. 

11.  Reduce  |f|.  12.  Reduce  ££*£. 

CASE  II. — Reducing  improper  fractions  to  whole  or 
mixed  numbers. 

13.  Reduce  y  to  a  whole  or  mixed  number. 

Suggestion. — The  value   of  a  fraction  is  Operation. 

the  quotient  of  the  numerator  divided  by  ^Q 

the  denominator.     We  therefore  divide  the  —     . 
numerator  by  the  denominator.     Hence, 

9O.  To  reduce  an  improper  fraction  to  a  whole  or 
mixed  number. 

Divide  the  numerator  by  the  denominator,  and  the 
quotient  will  be  the  whole  or  mixed  number  required. 

OBS.  The  remainder,  placed  over  the  divisor,  forms  a  part  of  the 
quotient,  and  must,  therefore,  be  annexed  to  the  integral  figures. 

14.  Reduce  '/  to  a  whole  or  mixed  number.   Ans.  3|. 

15.  Reduce  V-  16-  Reduce  3/- 
17.  Reduce  y.                              18.  Reduce  y. 
19.  Reduce  5-f.                              20.  Reduce  ff. 
21.  Reduce  Vk8.                            22.  Reduce  y?6- 
23.  Reduce  Jff.                            24.  Reduce  y//. 

CASE  III.  —  Reducing  mixed  numbers  to  improper 
fractions. 

25.  Reduce  the  mixed  number  13|  to  an  improper 
fraction. 

QUEST. — 90.  How  reduce  an  improper  frac.  to  a  whole  or  mixed  number  ? 


ARTS.  90,  91.]         COMMON  MULTIPLE.  71 

Suggestion. — Since  in  1  unit  there  are  3  Operation. 

thirds,  in  13,  there  are  13  times  as  many.  13| 

We  therefore  reduce  the   13  to  thirds,  by  3 

multiplying  it  by  3,  because  3  thirds  make  •  4f 

a  whole  one ;  and  adding  the  2  thirds,  we  ~3~Ans. 
have  41  thirds.    Hence, 

9 1 .  To  reduce  a  mixed  number  to  an  improper  frac- 
tion. 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  to  the  product  add  the  given  numerator. 
The  sum  placed  over  the  given  denominator,  will  form  the 
improper  fraction  required. 

OBS.  1.  A  whole  number  may  be  expressed  in  the  form  of  a  frac- 
tion without  altering  its  value,  by  making  1  the  denominator. 

2.  A  whole  number  may  also  be  reduced  to  a  fraction  of  any 
denominator,  by  multiplying  the  given  number  by  the  proposed 
denominator ;  the  product  will  be  the  numerator  of  the  fraction 
required.  Thus,  25  may  be  expressed  by  -2T5-,  ±J-^  or  YB°J  &c- 

26.  Eeduce  7f  to  an  improper  fraction.  Ans.  3J. 

27.  Reduce  8|.                       28.  Eeduce  14|. 
29.  Reduce  25f.                     30.  Reduce  30£. 
31.  Reduce  43T7T.                   32.  Reduce  61T4g. 
33.  Reduce  108*..                 34.  Reduce  210}£. 

35.  Reduce  63  to  4ths.  36.  Reduce  225  to  llths. 
CASE  IV.  Reducing  compound  fractions  to  simple  ones. 
37.  Reduce  f  of  J  to  a  simple  fraction. 

Suqqestion. — Multiplying  the  two          ^ 

r  J  Operation. 

numerators  together,  and  the  two  de- 
nominators, we  have  ^  ;  and  ,£    re-       txJ  —  r2- 
duced  to  its  lowest  terms  is  equal  to     anc*  i~2.= 
|,  which  is  the  answer  required.     Hence, 

QUEST.— 91.  HOW  reduce  a  mixed  number  to  an  improper  fraction  ?  Obs. 
How  express  a  whole  number  in  the  form  of  a  fraction  ?  How  reduce  a 
whole  number  to  a  fraction  of  a  given  denominator  ? 


72  FRACTIONS.  [SECT.  VI. 

92.  To  reduce  compound  fractions  to  simple  ones. 

Multiply  all  the  numerators  together  for  a  new  nume- 
rator, and  all  the  denominators  for  a  new  denominator. 

OBS.  If  the  example  contains  whole  or  mixed  numbers,  they  must 
be  reduced  to  improper  fractions,  before  multiplying. 

38.  Reduce  f  of  |  of  21  of  4  to  an  improper  fraction. 
Solution. — The    expression    21   of  4=f   of   j- ;    and 

f  xjx£  xf  =  HH,  orf£.     Ans. 

39.  Reduce  1  of  j  of  f         40.  Reduce  1  of  f  of  if  of  5. 
41.  Reduce  \  of  if  of  T6T.    42.  Reduce  £  of  W  of  f  of  3^. 
43.  Reduce  }f  of  ff  of  if.  44.  Reduce  -J  of  41  of  45. 

45.  Reduce  5f  of /T  of  ^  of  17. 

46.  Reduce  if  of  ¥9T  of  2i  of  7|. 

Contraction  by  Cancellation. 

47.  Reduce  1  of  |  of  f  of  -f  to  a  simple  fraction. 

Suggestion. — Since  the  product  Operation 

of  the  numerators  is  to  be  divided     i       ^      $      5      5 
by  the  product  of  the,,  denominators,     ^  of  —  of-  of  -- = — 
we  cancel  the  factors  2  and  3,  which 
are  common  to  both.     This  divides  the  terms  of  the  new 
fraction  by   the  same  number,  and  therefore  does  not 
alter  its  value.     Then  multiplying  the  remaining  factors 
together,  we  have  -/j,  the  answer  required.     Hence, 

93.  To  reduce  compound  fractions  to  simple  ones  by 
CANCELLATION. 

Cancel  all  the  factors  common  to  the  numerators  and 
denominators ;  then  multiply  the  remaining  factors  to- 
gether as  before. 

48.  Reduce  f  of  -|  of  f  to  a  simple  fraction.     Ans.  z. 

49.  Reduce  f  of  f  of  4.  50.  Reduce  f  of  |  of  f  of  1. 
51.  Reducef  of  |  of  f  of  li.  52-  Reduce-f  of  T2Fof  T7Tof  3. 


ARTS,  92 — 95.1  FRACTIONS.  73 

53.  ReduceTTFof  }£of  f  of  5.  54.  Reduce  ff  of  ff  of  ij. 
55.  Reduce  tf  of  if  of  f  f     56.  Reduce  jf  of  f  f  of  jf  f . 

57.  Reduce  if  of  46  of  29*. 

58.  Reduce  ||  of  |-|  of  TY<r  of  48. 

59.  Reduce  f  £  of  f  |  of  J|  of  84. 

60.  Reduce  21  off  of  TYof  110. 

Note. — For  method  of  reducing  Complex  Fractions  to  Simple 
ones,  see  Art.  143. 

CASE  Y. — Reducing  fractions  to  a  common  denominator. 

94,  Two  or  more  fractions  have  a  common  denomi- 
nator, when  they  have  the  same  denominator. 

61.  Reduce  |>  f,  and  j  to  a  common  denominator. 

Suggestion. — After  each  n  urn  era-  ^ 

...       _  .  Operation. 

tor,  we  write  all  the  denominators 

•  •  •  llxHx5tf> 

of  the   given   fractions,,    except   its  _  — — ^ 

own.  with  the  si  an  X  between  them,     2     2x3x5 

i            i,  r         i           n  ,T      T         2     2  X  2  X  5      20 
and  under  each  line  place  all  the  de-     -— r= . 

nominators  in  like  manner.     Then 

/.        .       .T  IA«   i-     >•         •    T       33x2x3      US 

performing;  the  multiplications  indi-     -  — =  — . 

*>      2x3x5      30 
cated,  the  products  are  equal  to  the 

given  fractions,  and  are  the  answer  required.     Hence, 

95.  To  reduce  fractions  to  a  common  denominator. 

Multiply  each  numerator  into  all  the  denominators  ex- 
cept its  owntfor  a  new  numerator,  and  all  the  denomina- 
tors, together  for  a  common  denominator. 

OBS. —  Compound  fractions  must  be  reduced  to  simple  ones,  and 
mixed  numbers  to  improper  fractions,  before  attempting  to  reduce 
them  to  a  common  denominator. 

62.  Reduce  J-  of  f,  2£,  and   2  to  a  com.  denominator. 

Suggestion. — i  of  f =f>  2|  =  f,  and  2  =  f.  Reducing 
f ,  |,  and  f  to  a  common  denominator,  they  Lecome  ¥\, 
||,  and  f|.  Ans. 

QUEST.   05. — How  reduce  fractions  to  a  common  denominator? 


74  FRACTIONS.  [SECT.  VI. 

Reduce  the  following  to  a  common  denominator. 
63.  Reduce  f,  f ,  and  1.          64.  Reduce  f,  f,  and  if. 
65.  Reduce  f ,  4,  and  T9r.        66.  Reduce  r\y  f  1,  and  f  f . 
67.  Reduce  if,  f  1,  and  }f  £.  68.  Reduce  Ji,  f  f,  and  ^V 
69.  Reduce  T5F,f,  and  i  of  17.  70.  Reduce  f,  41,  and  ff. 

CASE  VI. — Reducing  fractions  to  their  least  common  de- 
nominator. 

71.  Reduce  f,  f ,  and  f  to  the  least  com.  denominator. 

Suggestion.- We  first  find    the  Operation. 

least' common  multiple  of  all  the  given  n 

denominators,  which  is  24  ;  and  this 

O\o  ffQffA 

is  the  least  common  denominator  re-  ' 

quired.     The  next  step  is  to  reduce     rt 

/,        -  ,      ,,       2x2x3x2  =  24 

the  given  fractions  to  twenty-fourths 

without  altering  their  value.  The  denominator  of  the 
first  fraction,  is  contained  in  24,  6  times ;  and  multiplying 
both  terms  of  the  fraction  J  by  6,  it  becomes  if.  The 
denominator  6  is  contained  in  24,  4  times  ;  and  multi- 
plying the  fraction  f  by  4,  it  becomes  ^-.  The  denomi- 
nator 8  is  contained  in  24,  3  times ;  and  multiplying  the 
fraction  f  by  3,  it  becomes  if.  Now  if,  ^  and  if-  are 
equal  to  the  given  fractions  f ,  f ,  and  f .  Hence, 

96,  To  reduce  fractions  to  their  least  common  de- 
nominator. 

I.  Find  the  least  common  multiple  of  all  the  denomi- 
nators of  the  given  fractions,  and  it  will  be  the  least  com- 
mon denominator. 

II.  Divide  the  least  common  denominator  by  the  de- 
nominator of  each  of  the  given  fractions,  and  multiply 
the  numerator  by  the  quotient ;  the  products  will  be  the 
numerators  required. 

QUEST.   9G. — How  reduce  fractions  to  the  loast  common  denominator. 


ART.  96.]  FRACTIONS.  75 

OBS.  —  If  the  example  contains  compound  fractions,  whole  or 
mixed  numbers,  they  must  first  be  reduced  to  simple  fractions,  then 
all  must  be  reduced  to  their  lowest  terms  ;  otherwise  the  least  com- 
mon multiple  of  their  denominators,  may  not  be  the  least  common 
denominator. 

72.  Reduce  f,  f  ,  f  to  the  least  common  denominator. 

Suggestion.  —  First  find  the  least  common  multiple  of  the 

denominators,  which  is  2  x  3  x  2  =  12.   Now       Operation. 

12-4-3  =  4,  and  4  x  2  =  8,  the  1st  numerator.      2)3"4"6 


"  3)3"2"3 

5  =  10,  the  3d       "   __  1"2"1 


T82  7  T92,  and  |f. 
Reduce  the  following  to  the  least  common  denominator. 
73.  Reduce  J,  £,  and  f  .          74.  Reduce  f  ,  £,  and  T\. 
75.  Reduce  §,  f,  and  TV        76.  Reduce  f,  £,  },  and  fy. 
77.  Reduce  ^,  T9j,  and  11.    78.  Reduce  f  },-&,$,  and  T\. 
79.  Reduce  J  j,  |i,  and  /¥.     80.  Reduce  1^,  f,  1,  and  10. 
81.  Reduce  f  of  14,  and  3f  82.  Reduce  f  £,  4|,  and  27. 
83.  Reduce  if  of  T\,  and  16|.  84.  Reduce  |  of  2  1,  and  4  JL. 

85.  Reduce  f  ,  T7^,  }j,  and  -^  to  the  common  denomi- 

nator 120. 

86.  Reduce  f  ,  T^,  -r<b  anc^  A  *°  *ne  common  denomi- 

nator 144. 

87.  Reduce  -Jf  ,  |i,  and  f  |  to  the  common  denomi- 

nator 1080. 

88.  Reduce  f,  }|,  and  T7g-  to  thirty  sixths. 

89.  Reduce  T8T,  f  i,  and  |y  to  one  hundred  and  seconds. 

90.  Reduce  if,  if,  and  1  of  f  to  hundredths. 

91.  Reduce  f  of  1,  |J,  and  f|  to  ninety-sixths. 

92.  Reduce  1,  f  ,  and  T7^  to  twenty-fourths. 

QUKST.  —  of)s.  When  the  example  contains  compound  fractions,  whole  01 
mixed  numbers,  how  proceed  ? 


76  ADDITION    OF    FRACTIONS.  [SECT.  VI. 


ADDITION  OF  FRACTIONS. 

97.  If  two  or  more  fractions  have  a  common  denom- 
inator, the  parts  of  a  unit  expressed  by  their  numerators, 
are  of  -the   same   value  or  denomination,  and,  therefore, 
may  be  added  like  whole  numbers. 

Ex.  1.  A  man  gave  f  of  a  dollar  to  one  child,  f  to  an- 
other, and  f  to  another  ;  how  much  did  he  give  to  all  ? 

Suggestion.—  Add    the    mi-  Operation. 

merators,    and    place   the    sum     r^f- +*._-«    Or  Ud. 
over  the  common  denominator. 

Thus   2   eighths  and  3  eighths  are  5  eighths,  and  5  are 
10  eighths.     But   V0-  —  1!?  or  U  dollar. 

2.  Add  4,  f ,  and  f  3.  Add  f ,  f ,  and  f . 

4.  Add  TV,  W,  and  TV  5.  Add  T^,  |i,  and  |J. 

6.  A  man  bought  \  a  barrel  of  flour  at  one  time,  f  of 
a  barrel  at  another,  and  J  of  a  barrel  at  another ;  how 
much  did  he  buy  in  all  ? 

CY  ,.  -D    T  Operation. 

Suggestion.  —  Reduce 

the   fractions  to   a    com-  1  X  3  X  4  =  12,  1st  numerator 

,  2x2x4  =  16,  2d         " 

mon     denominator,     and  g^xS^lS^d         « 

add  the  numerators.    The  2  x  3  x  4  =  24,  com.  denom. 
result  if  =  Iff,   or    l}i.  Ans.  |f,  or  l|i  bar. 

98.  From  these  illustrations,  we  deduce  the  following 

RULE  FOR  ADDITION  OF  FRACTIONS. 

Reduce  the  fractions  to  a  common  denominator  ;  add 
their  numerators,  and  place  the  sum  over  the  common  de- 
nominator. 


QURST.— 98.  "What  is  the  rule  for  addition  of  fractions?     07)8.  What  must 
bo  done  with  compound  fractions,  whole  and  mixed  numbers? 


ARTS.  97,  98.]       ADDITION  OF  FRACTIONS.  *7 

OBS.  1.  Compound  fractions  must  be  reduced  to  simple  ones, 
whole  and  mixed  numbers  to  improper  fractions,  and  all  of  them  to 
a  common  denominator,  then  add  them  as  above. 

2.  In  adding  mixed  numbers,  it  is  generally  more  convenient  to 
add  the  whole  numbers  and  fractional  parts  separately,  and  then 
unite  their  sums. 

3.  The  operation  may  frequently  be  shortened  by  reducing  the 
given  fractions  to  their  least  common  denominator,  and  adding  their 
numerators. 

7.  What  is  the  sum  of  1  of  f  ,  2i,  and  7  ? 

Suggestion.—  ±  of  f-  =  f  ,  2  1  =  f  ,  and  7  =  f  Operation. 
Now  reducing  f,  f,  and  J  to  a  common  de-        3L  =  JL 
iiominator  as  in  the  margin,  and  adding  their        j  —  IA. 
numerators,  the  result  is  223T°,  which  is  equal        |=  Y?8- 

~ 


to  9i±,  or  9TV  Ans.~9ii,  or  9TV 

8.  Add  |  and  f  .  9.  Add  f,  1,  and  J. 

10.  Add  A,  |,  and  f  .  11.  Add  f,  ^,  and  f  . 

12.  Add  |i,  if,  and  f.  13.  Add  T\,  f,  and  if. 

14.  Add  JL-,  f,  and  if  15.  Add  ||,  11,  and  ^f, 

16.  Add  f  of  f,  |,  and  Yf  17.  Add  4|,  5f  and  I7f 

18.  Add  I75f,  20f  and  43. 

19.  Add  f  of  45,  |f  off,  and  51. 

20.  A  grocer   bought  3  chests  of  tea,  one   containing 
125f   pounds,  another  95J   pounds,  and  the   other  113£ 
pounds  :  how  much  did  he  buy  ? 

21.  A  man  gave  25J  dolls,  for  a  cow,  7of  dolls,  for  a 
horse,  11  0|-  dolls,  for  a  buggy,  and  86T9^-  dolls,  for  a  har- 
ness :  how  much  did  he  pay  for  all  ? 

22.  If  you  spend  151  f  dolls,  for  clothing,  270/7  dolls. 
for  board,  and  83£  dolls,  for  traveling,  what  is  the  amount 
of  your  expenses  ? 

23.  The   smaller  of  two  numbers  is   251  T\,  and  the 
difference    between    them    is     135T2j  :     what      is      the 
greater  number  ? 


78  SUBTRACTION    OF    FRACTIONS.  SECT.  VI. 


SUBTRACTION  OF  FRACTIONS. 

99.  When  two  fractions  Lave  a  common  denomina* 
tor,  the  less  numerator  may  be  subtracted  from  the  greater, 
as  in  whole  numbers,  and  the  result  placed  over  the  com- 
vwn  denominator,  will  be  the  difference  between  them. 

Ex.  1.  If  I  buy  ||  of  an  acre  of  land,  and  afterwards 
sell  if  of  an  acre,  how  much  shall  I  have  left  ? 

Suggestion. — Taking  1 9  forty -fifths  from 
27  forty-fifths,  the  remainder  is  ^  of  an 
acre.  — nr 

2.  From  f  1  take  T\.  3.  From  if  take  T8T. 

4.  From  f  f  take  f  f .  5.  From  TVo  take  T4¥V 

6.  From  }  of  a  yard  of  cloth,  take  £  of  a  yard. 

Suggestion.  —  Since    these  Operation. 

fractions  have  not  a  common  3  x^__|2  ) 

denominator,  it  is  plain  that  1x5^   5  [  mimei'ators- 

one  numerator  cannot  be  taken  5  x  4  =  20  com.  denom. 

from  the  other.   We,  therefore,  -J|  —  ^  =  /y  yd.  Ans. 
reduce  them  to  a  common  de- 
nominator, then  subtract  as  above. 

1OO»  From  these  illustrations,  we  deduce  the  fol- 
lowing 

RULE  FOR  SUBTRACTION  OF  FRACTIONS. 

Reduce  the  fractions  to  a  common  denominator  ;  sub- 
tract the  less  numerator  from  the  greater,  and  place  the 
remainder  over  the  common  denominator. 

ODS.  1.    Compound  fractions  must  be  reduced  to  simple  ones, 

QITKST. — 100.  What  is  the  rule  for  subtraction?  Ol>s.  What  must  be  done 
ivith  compound  fractions,  whole  and  mixed  numbers?  How  else  are  mixed 
numbers  subtracted '? 


ARTS.  99,  100.]     SUBTRACTION  OF  FRACTIONS.  79 

whole  and  mixed  numbers  to  improper  fractions,  and  all  of  them  to 
a  common  denominator,  as  in  addition. 

2.  In  subtracting  mixed  numbers,  it  is  sometimes  more  conve- 
nient to  take  the  fractional  part  of  the  less  from  the  fractional  part 
of  the  greater,  then  the  integral  part  of  the  less  from  that  of  the 
greater.     (See  Ex.  17.) 

3.  In  subtracting  a  proper  fraction  from  a  whole  number,  wa 
may  borrow  a  unit  arid  take  the  fraction  from  this,  then  diminish 
the  whole  number  by  1. 

7.  From  ji  take  f .  8.  From  f  take  f . 

9.  From  if  take  f .  10.  From  51  take  11. 

11.  From  fi  take  ff,  12.  From  11  take  11. 

13.  From  ||  take  ff.  14.  From  41  take  f  f . 

15.  From  T«JT  take  f  f.  16.  From  ffc  take  i|. 

17.  From  7£  take  2i. 

Suggestion.  —  Reducing         First   Operation. 

the  mixed  numbers  to  im-  7i  —  2-2,  and  2-2=rif 

proper  fractions,  then  to  a  2i  =  fr  and  f  —  f  j- 

common    denominator    12,  ^Z  —  ZT  —  ^  or~5J^  Ans. 

they    become   f|    and    \17. 

• 

Or,  reducing  the  fractional  parts  Second  Operation 
to  a  common  denominator  12,  then  1^  =  7^ 

subtract  the  numerator  of  the  less  ^t~L?JJF 

from  that  of  the  greater,  Ans.  5T^ 

18.  From  9^  take  31.  19.  From  llf  take  6f . 
20.  From  12£  take  8|.  21.  From  18f  take  lOf. 

22.  From  4  take  f.  Ans.  31. 

23.  From  9  take  21.  24,  From  lof  take  7. 

25.  From  f  of  1  take  1  of  |.     Ans.  ^  —  TL. 

26.  From  |  of  |  of  a  yard,  take  1  of  f  of  a  yard. 

27.  From  ^  of  40  pounds,  take  1  of  50  pounds. 

28.  From  \  of  15}  m!!cs,  lake  £  of  7  miles. 


80  MULTIPLICATION    OF    FRACTIONS,       [SECT,  VI. 

MULTIPLICATION  OF  FRACTIONS. 

1  0  1  •  Multiplying  by  a  fraction  is  taking  a  certain 
PORTION  of  the  multiplicand  as  many  times,  as  there  are 
like  2>ortions  of  a  unit  in  the  multiplier.  That  is, 

Multiplying  by  £,  is  taking  1  half  of  the  multiplicand 
once.  Thus,  6  x  £  =  3, 

Multiplying  by  i,  is  taking  1  third  of  the  multiplicand 
once.  Thus,  6  x  |-  =  2, 

Multiplying  by  f  ,  is  taking  1  third  of  the  multiplicand 
twice*  Thus,  6  x  f  =  4, 

Ex.  1.  If  1  quart  of  condensed  milk  is  worth  J  of  a 
dollar,  how  much  are  4  quarts  worth  I 

Analysis.  —  If  1  quart  is  worth  f      First  Operation. 
dollar,  4  quarts  are  worth  4  times  f     3x4     12 
•dollar,  and   4  times  f  are  «-,  or  1|        8     ~1P  °r  **' 


dollar.     That  is,  we  multiply  the  nu-          ^Ans.  If  dolls. 
merator  by  the  whole  number. 

Or,  we  may  divide  the  denomi-      ~ 

'         _      J.    .  .        .       Second  Operation 

nator  by  the  whole  number,  when  it 

can  be  done  without  a  remainder,     -  =-,  or  14-. 

8-i-4.*   Q 
for  dividing  the  denominator  multi- 

plies the  value  of  the  fraction.     Hence, 

1  O2.  To  multiply  a  FRACTION  by  a  whole  number. 

Multiply  the  numerator  of  the  fraction  by  the  whole 
number^  and  write  the  product  over  the  denominator, 

Or,  divide  the  denominator  by  the  whole  number,  when 
this  can  be  done  without  a  remainder, 

OBS,  A  fraction  is  multiplied  into  a  number  equal  to  its  denomi- 
nator by  canceling  the  denominator.  Thus,  *  x  7  =  4* 

2.  Multiply  f  by  7.  3.  Multiply  }J  by  9. 

4.  Multiply  JL  by  8.  5.  Multiply  ff  by  10. 


r.— 10L  What  is  meant  by  multiplying  by  a  fraction  ? 


ARTS.  101— 103.J  MULTIPLICATION  OF  FRACTIONS.  81 

6.  Multiply  292  b7  12-  ?•  Multiply  if  by  21. 

8.  Multiply  If  by  16.  9.  Multiply  f  f  by  23. 

10.  Multiply  TVo  by  31.        11.  Multiply  iff  by  83. 

12.  What    cost   5    yards    of    cloth,    at   21  dollars   a 
yard? 

Analysis. — Five    yards    will   cost  5          Operation. 
times    as    much    a    1    yard.     Now,   5  21 

times  i  are  f  =  2J;  5  times  2  are  10,  5 

and  21  are  121.     Hence,  Ans.  121  dolls. 

1O3*  To  multiply  a  mixed  number  by  a  whole  one. 

Multiply  the  fractional  part  and  the  whole  number 
separately,  and  unite  the  products. 

13.  Multiply  12|  by  5.          14.  Multiply  21]-  by  9. 
15.  Multiply  35J  by  7.  16.  Multiply  29|  by  10. 
17.  Multiply  45£  by  12.         18.  Multiply  53f  by  21. 
19.  What   cost  |  of  a  pound  of  coffee,  at  25  cents  a 

pound  ? 

Analysis. — Since  1  pound  costs  25  First  Operation. 
cents,  |  of  $  pound  will  cost  f  of  25          3)25  cents, 
cents.     Now,  1  third  of  25  is  81,  and  "fcl 

2  thirds   is  2  times  81,  or  16|  cents.  J2^ 

We  divide  the  whole  number  by  3,  Ans.  16  J  cents, 
and  multiply  the  quotient  by  2. 

Or,  we   may    first   multiply   the  Second  Operation. 
whole    number   by    2,  then    divide  25  cents, 

this  product  by  3  ;  for  1  third  of 
2  times  25  is  obviously  the  same  as          3)50 

2  times  1  third  of  it.     Hence,  Ans.  16|  cents. 


QUEST. — 102.  How  multiply  a  fraction  by  a  whole  number.  Obs.  How  is 
a  fraction  multiplied  by  a  number  equal  to  its  denominator?  108.  How  mul- 
tiply a  mixed  number  by  a  whole  one  ? 


82  MULTIPLICATION  OF  FRACTIONS.          [SECT.  VI. 

104.  To  multiply  a  whole  number  by  a,  fraction. 

Divide  the  whole  number  by  the  denominator,  and 
multiply  the  quotient  by  the  numerator. 

Or,  multiply  the  whole  number  by  the  numerator,  and 
divide  the  product  by  the  denominator. 

OBS.  "When  the  whole  number  cannot  be  divided  by  the  denoinir 
nator  without  a  remainder,  the  latter  method  in  generally  preferred. 

20.  Multiply  21  by  f .  21.  Multiply  19  by  4. 

22.  Multiply  31  by  f .  23.  Multiply  43  by  f . 

24.  Multiply  39  by  f  25.  Multiply  47  by  f . 

26.  Multiply  75  by  T9T.  27.  Multiply  91  by  fy. 

28.  "What  cost  2£  gallons  of  cider,  at  31  cents  a  gall.  ? 

Analysis.  —  If    1    gallon    cost    31  2)31  cents, 

cents,  2i  gallons  will  cost  2^  times  31  2J 

cents.     Now,  2  times  31  are  62  ;  and  62 

i  of  31    is   15J,  which  added  to  62?  15jr 

makes  77^  cents.     Hence,  Ans.  7 7^ "cents. 

105.  To  multiply  a  whole  by  a  mixed  number. 

Multiply  first  by  the  whole  number,  then  by  the  frac- 
tion, and  add  the  products  together. 

29.  Multiply  22  by  3|.  30.  Multiply  46  by  4J. 
31.  Multiply  58  by  9f.           32.  Multiply  65  by  llf . 

33.  At  |  of  a  dollar  a  pound,  what  will  f  of  a  pound 

of  tobacco  cost  ? 

Operation. 
Analyses.- Since    1    pound        x  w         ^ 

costs  f  of  a  dollar,  f  of  a  pound 

will  cost  |  of  f  doll.,  and  f  of  f  =  2if,  or  T3o-  dollar.  We 
multiply  the  numerators  together  and  the  denominators 
together,  and  the  result  -f^,  reduced  to  the  lowest  terms 
is  T37  of  a  dollar.  Hence, 

QUEST.— 104.  TIow  multiply  a  whole  nnraber  by  a  fraction?  105.  IIo\v 
multiply  a  whole  by  a  mixed  number : 


ARTS.  104  —  10*7.]  MULTIPLICATION  OF  FRACTIONS.         83 
1O6.  To  multiply  a,  fraction  by  &  fraction. 

Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  together  for  a  new  denominator. 

OBS.  —  "When  the  numerators  and  denominators  contain  common 
factors,  they  should  be  canceled. 

34.  What  is  the  product  of  J-  multiplied  by  }|  ? 

Suggestion.  —  Canceling  the  fac-  Operation. 

tors  4  and  5  which  are  common,     ^     10   2     2 
the  result  is  f  .  $  x  Ig'  3~3*          * 

35.  Multiply  f  by  f  .  36.  Multiply  £  by  f  . 
37.  Multiply  £  by  f  .  38.  Multiply  T\  by  fy. 
39.  Multiply  T8j  by  if.  40.  Multiply  /T  by  if. 
41.  Multiply  |f  by  |J.  42.  Multiply  f  f  by  f  f  . 

43.  What  is  the  product  of  3T\  by  21  ? 

Analysis.  —  The  mixed   number  Operation. 

3T2o  =H>  and  3i=f     Multiply  the     f  |  x  f  =  \V,  or  8. 
numerators  together,  and  the  denominators. 

Or,    cancel   the    common   factors,  8,£$r$£     $  _ 
then  multiply.     Hence,  &  50     jfc 


1O7.  To   multiply    a   m^eo?   number  by    a  mixed 
number. 

^Reduce  the  mixed  numbers  to  improper  fractions,  and 
proceed  as  in  multiplying  a  fraction  by  a  fraction. 

43.  Multiply  7j  by  5£.  44.  Multiply  6T47  by  7j. 

45.  Multiply  21|  by  16}.        46.  Multiply  85r5¥  by  24|. 
47.  Multiply  75f  by  42J.       48.  Multiply  91  f  by  63T6r. 

QUEST.  —  106.  How  is  a  fraction  multiplied  by  a  fraction?     107.  How  mul- 
tiply one  mixed  number  by  another  ? 


84  DIVISION    OF    FRACTIONS.  [SECT.  VL 

DIVISION  OF  FRACTIONS. 

Ex.  1.  If  2  pounds  of  butter  costs  f-  of  a  dollar,  what 
will  1  pound  cost? 

Analysis.— Since  2  pounds  costs  f  Operation. 

doll,  1   pound  will  cost  1  of  |  doll.,     izL?_^  J0n. 
which   is  |  doll.  ;  we  divide  the  nu-        ** 
merator  of  the  fraction  ±  by  the  whole  number  2,  and 
set  the  quotient  over  the  denominator. 

2.  If  3   quarts  of  chestnuts  cost  f  of  a  dollar,  what 
will  1  qtiart  cost  ? 

Analysis. — In   this  case   we  cannot         Operation. 
divide  the  numerator  2  by  3,  without  a        "    __"  c|o]j 
remainder.     We  therefore  multiply  the     3x3     9 
denominator  by  it,  which  divides  the  fraction.     Hence, 

1C  8.  To  divide  a  fraction  by  a  whole  number. 

Divide  the  numerator   by  the  whole  number,  when  it 
can  be  done  without  a  remainder.     If  not, 

Multiply  the  denominator  by  the  whole  number. 

3.  What  is  the  quotient  of  -f  divided  by  3  ?     Ans.  f . 

4.  Divide  }|  by  2.  5.  Divide  f-£  by  5. 
6.  Divide  |£  by  10.  7.  Divide  if  by  11. 
8.  Divide  TVo  by  15.  9.  Divide  iff  by  60. 

10.  At  i  of  a  penny  a  yard,  how  many  yards  of  tape 
can  you  buy  for  f  of  a  penny  ? 

Analysis.— If  £  of  a  penny  will  buy          Operation. 
1  yard,  f  of  a  penny  will  buy  as  many     l^l^3  yards, 
yards  as  1  fourth  is  contained  times  in  3  fourths,. which 
is  3.     That  is,  as  the  fractions  have  a  common  denomi- 
nator, we  divide  one  numerator  by  the  other. 

QUEST. — 108.  How  is  a  fraction  divided  by  a  whole  number? 


ARTS.  108,  109.]     DIVISION  OF  FRACTIONS.  85 

11.  At  |  of  a  dollar  a  yard,  how  much  silk  can  be 
bought  for  f  of  a  dollar? 

Analysis. — Since  these  fractions  ^         .. 

Operation. 
have    different   denominators,   we     Axl_12    or  -p     ^ 

invert  the  divisor,  and  proceed  as     5  J 

in  multiplication  of  fractions.     Hence, 

1O9.  To  divide  &  fraction  by  a  fraction. 

1.  If  the  given  fractions  have  a  common  denominator ', 
divide  the  numerator  of  the  dividend  by  the  numerator 
of  the  divisor. 

11.  When  the  fractions  have  not  a  common  denomina- 
tor, invert  the  divisor,  and  proceed  as  in  multiplication 
of  fractions. 

OBS.  1.    Compound  fractions  must  be  reduced  to  simple  ones, 
and  mixed  numbers  to  improper  fractions. 

2.  After  inverting  the  divisor,  the  factors  common  to  the  nu- 
merators and  denominators  should  be  canceled. 

12.  What  is  the  quotient  of  }£  divided  by  f  ? 

Suggestion. — Invert  the  divisor,     5,  10     J     5 
and  cancel  the  common  factors.          4,  I#     #~4' 

13.  Divide  if  by  f  14.  Divide  |f  by  i£. 
15.  Divide  |f  by  f  J.  16.  Divide  TW  by  f  f 

17.  Divide  f  of  |  of  51  by  f  of  ^-.  Ans.  f. 

18.  Divide  J  of  }f  by  f  off. 

19.  Divide  f  of  81  by  f  of  TV 

20.  Divide  4  of  28  by  |  of  f  of  5J. 

21.  Divide  f  of  T%  of  7^  by  -ff  of  40. 

22.  At  5i  dollars  a  yard,  how  much  cloth  can  be 
bought  for  26|  dolls.  *  Ans.  4J  yds. 

23.  Divide  221  by  6f .  24.  Divide  381  by  T3r. 
25.  Divide  15f  by  f  of  12.       26.  Divide  f  of  20  by  2f. 

QUEST. — 109.  How  is  one  fraction  divided  by  another  when  they  have  2 
commit)  'ionominator  ?    How,  when  t>?v  have  not  a  comiron  denominator  > 


86  DIVISION    OF   FRACTIONS.  [SECT.  VI. 

27.  At  £  of  a  dollar  a  pound,  how  many  pounds  of 
tea  will  25  dollars  buy  ? 

Analysis. — If  f  of  a  dollar  will  buy  1  Operation. 
pound,  25  dollars  will  buy  as  many  pounds  25 

as  J  is  contained  times  in  25.     We  multi-  4 

ply  the  whole  number  by  the  denominator,     3)100 
and  divide  the  product  by  the  numerator.    Ans.  33^  p. 

Or,  we  may  reduce  the  whole  number  to  the  form 
of  a  fraction,  and  then  divide  it  by  the  fraction.  Thus, 
25i=2T5,  and  y-^i  =  ¥  xf=J-fA,  or  33J  pounds. 

1 1O.  To  divide  a  whole  number  by  a,  fraction. 

Multiply  the  whole  number  by  the  denominator,  and 
divide  the  product  by  the  numerator. 

Or,  reduce  the  whole  number  to  the  form  of  a  fraction, 
and  proceed  according  to  the  rule  for  dividing  a  fraction 
by  a  fraction. 

OBS.  When  the  divisor  is  a  mixed  number,  it  must  be  reduced 
to  an  improper  fraction,  then  proceed  as  above. 

28.  Divide  115  by  f.  29.  Divide  147  byj. 
30.  Divide  180  by  }i.  31.  Divide  218  by  Jfc. 
32.  Divide  228  by  15J.          33.  Divide  360  by  20f. 

3 

34.  Reduce  the  complex  fraction  ^  to  a  simple  one. 

"3" 

Analysis. — The   denominator    of   a          f 3.0 

fraction  is  a  divisor;  hence  the  opera-          |     T 
tion  is  the  same  as  dividing!  by  f.     f  xf =£,  or  1J-. 
But  to  divide  J  by  f ,  we  invert  the  divisor,  then  multiply 
the  numerators  together  and  the  denominators.     Hence., 

111.  To  reduce  a  complex  fraction  to  a  simple  one. 

Consider  the  denominator  as  a  divisor,  and  proceed  as 
in  division  of  fractions. 

QUEST.— 110.  How  is  a  whole  number  divided  by  a  fraction^ 


ARTS.  110,  111.]      COMPLEX  FRACTIONS.  87 

OBS.  When  complex  fractions  are  reduced  to  simple  ones,  they 
are  added,  subtracted,  multiplied,  and  divided,  according  to  the  pre-! 
ceding  rules. 


5— 
35.  Reduce-1. 

23^ 

38.  Eeduce  f. 

9" 

36.  Eeduce  —  -. 

34 

02 
39.  Reduce  -|. 

37.  Reduce  -A 

25 
40.  Reduce  —  . 

TO 

EXERCISES    IN    FRACTIONS. 

1.  If  a  man  can  earn  T9F  of  a  dollar  per  day,  now 
much  can  he  earn  in  J  of  a  day? 

2.  If  water  runs  3|  miles  per  hour,  how  far  will  it  run 
in*  121  hours? 

3.  What  cost  48f  barrels  of  beef,  at  16f  dolls,  a  barrel  ? 

4.  How  far  can  a  man  travel  in  851  days,  if  he  travels 
33J  miles  per  da}7? 

5.  What  cost  581  cheeses,  at  4J  dollars  apiece  ? 

6.  What  cost  85-2vpounds  of  sugar,  at  7£  cents  a  pound? 

7.  What  cost  67f  bu.  of  barley,  at  65^  cents  a  bu.  ? 

8.  A  grocer  sold  £  of  56  gallons  of  molasses,  at  1  of 
3  dollars  a  gallon  :  how  much  did  it  come  to  ? 

9.  If  I  pay  i  of  -|  of  28  dollars  for  a  barrel  of  flour, 
how  much  must  I  pay  for  |  of  3f  barrels? 

10.  A  grocer  sold  some  figs  for  if  of  a  dollar,  which 
was  T2T  of  a  dollar  a  pound :  how  many  did  he  sell  ? 

11.  At  61   cents  apiece,  how  many  oranges  can  be 
bought  for  621  cents? 

12.  How  much  butter,  at  18f  cents,  a  pound,  can  be 
bought  for  951  cents? 

13.  A  man  traveled  125|  miles  in  7f  days  :  how  far 
did  he  travel  per  day? 

14.  A  lad  having  871  cents,  spent  it  in  candy,  at  371 
cents  a  pound  :  how  much  did  he  buy  ? 

15.  A  man  spent  f  of  720  dollars  for  tobacco,  which 
was  |  of  |  of  a  dollar  a  pound  :  how  much  did  he  get1? 


88  COMPOUND  .  [SECT.  VII. 

SECTION  VII. 
COMPOUND  OR  DENOMINATE  NUMBERS. 

ART.  87.  SIMPLE  Numbers  are  those  which  express 
units  of  the  same  kind  or  denomination ;  as,  one,  two, 
three  ;  4  pears,  5  feet,  &c. 

COMPOUND  Numbers  are  those  which  express  units 
of  different  kinds  or  denominations  ;  as  the  divisions  of 
money,  weight,  and  measure.  Thus,  6  shillings  and  7 
pence ;  3  feet  and  7  inches,  <fec.,  are  compound  numbers. 

Note. — Compound  Numbers  are  sometimes  called  Denominate 
Numbers. 

FEDERAL  MONEY. 

88«  Federal  Money  is  the  currency  of  the  United 
States.  Its  denominations  are,  Eagles,  dollars,  dimes, 
cents,  and  mills. 

10  mills  (m.)  make  1  cent,     marked  ct. 

10  cents  "      1  clime,         "       d. 

10  dimes  "     1  dollar,        "       doll,  or  $. 

10  dollars  "     1  eagle,         "       E. 

89«  The  national  coins  of  the  United  States  are  of 
three  kinds,  viz  :  gold,  silver,  and  copper. 

1.  The  gold  coins  are  the  eagle,  half  eagle,  and  quarter 
eagle,  the  double  eagle*  and  gold  dollar* 

2.  The  silver  coins  are  the  dollar,  half  dollar,  quarter 
dollar,  the  dime,  half  dime,  and  three-cent-piece. 

QUKST. — 87.  What  are  simple  numbers  1  What  are  compound  numbers  ? 
feS.  What  is  Federal  Money  1  Recite  the  Table.  89.  Of  how  many  kinds  are 
the  coins  of  the  United  States?  What  are  the  gold  coins  ?  What  are  the 
fciiver  coins  ? 

Added  by  Act  of  CoKg»«*«s,  Feb.  20th,  1849. 


ARTS.  87 — 91.1          .NUMBERS.  89 

3.  The  copper  coins  are  the  cent  and  half  cent. 
Mills  are  not  coined. 

Obs.  Federal  money  was  established  by  Congress,  August  8th, 
1786.  Previous  to  this,  English  or  Sterling  money  was  the  princi- 
pal currency  of  the  country. 

STERLING  MONEY. 

90.  English  or  Sterling  Money  is  the  national  cur- 
rency of  Great  Britain. 

4  farthings  (qr.  or  far.)  make  1  penny,  marked         d. 
12  pence  "      1  shilling,     "  s. 

20  shillings  '       1  pound  or  sovereign,  £. 

21  shillings  "       1  guinea. 

OBS.  The  Pound  Sterling  is  represented  by  a  gold  coin,  called 
a  Sovereign.  Its  legal  value,  according  to  Act  of  Congress,  1842,  \* 
$4.84 ;  its  intrinsic  value,  according  to  assays  at  the  U.  S.  mint,  is 
$4.851.  The  legal  value  of  an  English  shilling  is  24-1  cents. 

TROY  WEIGHT. 

91.  Troy   Weight  is  used  in  weighing  gold,  silver, 
jewels,  liquors,  &c.,  and  is  generally  adopted  in  philo- 
sophical experiments. 

24  grains  (gr.)        make  1  pennyweight,  marked  piut. 
20  pennyweights        "     1  ounce,  "       oz. 

12  ounces  "     1  pound,  "       lb. 

Note. — Most  children  have  very  erroneous  or  indistinct  ideas  of 
the  weights  and  measures  in  common  use.  It  is,  therefore,  strongly 
recommended  for  teachers  to  illustrate  them  practically,  by  referring 
to  some  visible  object  of  equal  magnitude,  or  by  exhibiting  the  ounce, 
the  pound  ;  the  linear  inch,  foot,  yard,  and  rod ;  also  a  square  and 
cubic  inch,  foot,  &c. 

QUEST.— What  are  the  copper  coins  ?  Obs.  When  and  by  whom  was  Federal 
Money  established  ?  90.  What  is  Sterling  Money  ?  Repeat  the  Table.  Vbs.  By 
what  is  the  Pound  Sterling  represented  ?  What  is  its  legal  value  in  dollars  ant! 
cents?  What  is  the  value  of  an  English  shilling?  01.  li>  what  is  Troy  Weight 
used?  Rer^e  tho  Table. 

4 


90  COMPOUND  [SECT.  VII. 

AVOIRDUPOIS  WEIGHT. 

92.  Avoirdupois  Weiyht  is  used  in  weighing  groceries 
and  all  coarse  articles  ;  as  sugar,  tea,  coffee,  butter,  cheese, 
flour,  hay,  &c.,  and  all  metals  except  gold  and  silver. 
16  drams  (dr.)  make  1  ounce,  marked     oz. 

16  ounces  "     1  pound,       "  Ib. 

25  pounds  "     1  quarter,     "  qr. 

4  quarters,  or  100  Ibs.    "     1  hundred  weight,  cwt. 
20  hund.,  or  2000  Ibs.      "     1  ton,  marked          T. 
OBS.  1.  Gross  weight  is  the  weight  of  goods  with  the  boxes,  casks, 
or  bags  which  contain  them,  and  allows  112  Ibs.  for  a  hundred  weight 
Net  weight  is  the  weight  of  the  goods  only. 

2.  Formerly  it  was  the  custom  to  allow  112  pounds  fora  hundred 
weight,  and  28  pounds  for  a  quarter ;  but  this  pract«ce  has  become 
nearly  or  quite  obsolete.  The  laws  of  most  of  the  states,  as  well  as 
general  usage,  call  100  Ibs.  a  hundred  weight,  and  25  Ibs.  a  quarter. 
In  estimating  duties,  and  weighing  a  few  coarse  articles,  as  iron, 
dye-woods,  and  coal  at  the  mines,  112  Ibs.  are  still  allowed  for  a 
hundred  weight.  Coal,  however,  is  sold  in  cities,  at  100  Ibs.  for  a 
hundred  weight. 

APOTHECARIES'  WEIGHT. 

93»  Apothecaries'  Weight  is  used  by  apothecaries  and 
physicians  in  mixing  medicines. 

20  grains  (yr.)  make  1  scruple,  marked   sc.  or  3. 
3  scruples  "      1  dram,  "        dr.  or  3. 

8  drams  "      1  ounce,         "        oz.  or  g. 

12  ounces  "      1  pound,        "  ft. 

OBS.  1.  The  pound  and  ounce  in  this  weight  are  the  same  as  the 
1\oy  pound  and  ounce;  the  subdivisions  of  the  ounce  are  different. 
2.  Drugs   and  medicines  are  bought   and  sold  by  avoirdupois 
weight. 

QUKST.— 92.  In  what  is  Avoirdupois  Weight  used  ?  Recite  the  Table.  Obs. 
What  is  gross  weight  ?  What  is  net  weight  ?  How  many  pounds  were  for- 
merly allowed  for  a  quarter?  How  many  for  a  hundred  weight?  93.  In  what 
is  Apothecaries  Weight  used  ?  Repeat  the  Table.  Obs.  To  what  are  the  Apo- 
thecaries' pound  and  ounce  equal?  How  are  drugs  and  medicines  bought 
and  sold  ? 


A.RTS.  92 95.]  JV UMBERS.  91 

LONG  MEASURE. 

94.  Long  Measure  is  used  in  measuring  length  or 
distances  only,  without  regard  to  breadth  or  depth. 

12  inches  (in.)                make  1  foot,  marked  ft. 

3  feet                               "  1  yard,  "       yd. 

5J  yards,  o*  16£  feet        "  1  rod,  perch,  or  pole,  r.  or  p. 

40  rods                               "  1  furlong,  marked  fur. 

8  furlongs,  or  320  rods    "  1  mile,  "       m. 

3  miles                             "  1  league,  "       I. 
60  geographical  miles,  or  > 

>  t(       1    norrrt»A  '*       a  a  ft    nr  ° 

69i  statute  miles  $          deSree' 

360  deg.  make  a  great  circle,  or  the  circum.  of  the  earth. 

OES.  1.  4  inches  make  a  hand;  9  inches,  1  span;  18  inches,  1 
cubit ;  6  feet,  1  fathom ;  4  rods,  1  chain ;  25  links,  1  rod. 

2.  Long  measure  is  frequently  called  linear  or  lineal  measure. 
Formerly  the  inch  was  divided  into  3  barleycorns  ;  but  the  barley- 
corn, as  a  measure,  has  become  obsolete.  The  inch  is  commonly 
divided  either  into  eighths,  or  tenths;  sometimes  it  is  divided  into 
twelfths,  which  are  called  lines.  , 

CLOTH  MEASURE. 

95»  Cloth  Measure  is  used  in  measuring  cloth,  lace,  and 
all  kinds  of  goods,  which  are  bought  or  sold  by  the  yard. 

2j  inches  (in.)  make  1  nail,  marked  na. 

4  nails,  or  9  in.      "     1  quarter  of  a  yard,       "      qr. 

4  quarters  "     1  yard,  "      yd. 

'  3  quarters  "     1  Flemish  all,  "      Fl.  c> 

5  quarters  "     1  English  elh  "      E.  e. 

6  quarters  "     1  French  eli,  "      F.  e. 


QUEST.— 94.  In  what  is  Long  Measure  used?  Repeat  the  Table.  Draw  a 
line  an  inch  long  upon  your  slate  or  black-board.  Draw  one  two  inches  long. 
Draw  another  a  foot  long.  Draw  one  a  yard  long.  How  long  is  your  teacher's 
desk  ?  How  long  is  the  school-room  ?  How  wide  ?  Obs.  What  is  Long  Meas- 
ure frequently  called  ?  Flow  is  the  inch  commonly  divided  at  the  present 
d.-iy  ?  95.  In  what  is  Cloth  Measure  used  ?  Repeat  the  Table. 


92 


COMPOUND 


[SECT.  VIL 


1  sq.  rod,  perch, 
or  pole, 
1  rood, 

sq.  r. 

R. 

1  acre, 

A. 

1  square  mile,      " 

M. 

OBS.  Cloth  measure  is  a  species  of  long  measure.  The  yard  is 
the  same  in  both.  Cloths,  laces,  (fee.,  are  bought  and  sold  by  the 
linear  yard,  without  regard  to  their  width. 

SQUARE  MEASURE. 

96.  Square  Measure  is  used  in  measuring  surfaces, 
or  things  whose  length  and  breadth  are  considered  with- 
out regard  to  height  or  depth  ;  as  land,  flooring,  plaster- 
ing, &c. 

144  square  in.  (sq.  in.)  make  1  square  foot,  marked  sq.ft. 
9  square  feet  "       1  square  yard,      "       sq.  yd* 

30|  square  yards,  or  )     (     ( 
2  7  2|  square  feet          f          ( 
40  square  rods  " 

4  roods,  or  160  sq.rds.  " 
640  acres  " 

OBS.  1 .  A  square  is  a  figure,  which  has  four  equal  sides,  and  all 
its  angles  right  angles,  as  seen  in  the  adjoining  diagram.  Hence, 

2.  A  Square  Inch  is  a  square,  whose  sides         9  sq.ft.=l  sq.  yd. 
are  each  a  linear  inch  in  length. 

A  Square  Foot  is  a  square,  whose  sides 
are  each  a  linear  foot  in  length. 

A  Square  Yard  is  a  square,  whose  sides 
are  each  a  linear  yard  or  three  linear  feet  in 
length,  and  contains  9  square  feet,  as  re- 
presented in  the  adjacent  figure. 

3.  In  measuring  land,  surveyors   use  a 
chain  which  is  4  rods  long,  and  is  divided 

into  100  links.  Hence,  25  links  make  1  rod,  and  7-f-fa  inches 
make  1  link. 

This  chain  is  commonly  called  Gunter's  Chain,  from  the  name 
of  its  inventor. 

4  Square  ^Measure  is  sometimes  called  Land  Measure,  because 
it  is  used  in  measuring  land. 

QI-KST.— Obs.  Of  what  is  Cloth  Measure  a  species?  96.  In  what  is  Square 
Measure  used  ?  Repeat  the  Table.  Obs.  What  is  a  square  ?  What  is  a  square 
inch?  What  is  a  square  foot?  A  square  yard?  Can  you  draw  a  square 
Inch  '(  Can  you  draw  a  square  foot?  A  ?qmire  yard  ? 


ARTS.  9G,  97.] 


NUMBERS. 


CUBIC   MEASURE. 

97  •   Cubic  Measure  is  used  in  measuring  solid  bodies, 
or   things   which    have    lenglht    breadth,    and    thick  nests; 
such  as  timber,  stone,  boxes  of  goods,   the  capacity  of ' 
rooms,  <fec. 
1728  cubic  inches  (cu.  in.)  make  1  cubic  foot,    marked  cu.ft. 


2  7  cubic  feet 

1  cubic  yard,        " 

cu.  yd. 

40  feet  of  round,  or     >     tt 

1  ton,                     " 

T. 

50  ft.  of  hewn  timber,  > 

42  cubic  feet 

1  ton  of  shipping,  " 

T. 

16  cubic  feet 

(  1  cord  foot,  or  a  (f 
(     foot  of  wood, 

c.ft. 

8  cord  feet,  or  >              (( 
28  cubic  feet      ) 

1  cord, 

C. 

OBS.  1.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high, 
contains  1  cord.  For  8  into  4  into  4=1*28.  27  cu.  ft.=l  cu.  yd. 

2.  A  Cube  is  a  solid  body  bounded  by 
six  equal  squares.     It  is  often  called  a  licx- 
aedron.     Hence, 

A  Cubic  Inch  is  a  cube,  each  of  whose 
sides  is  a  square  inch,  as  represented  by 
the  adjoining  figure. 

A  Cubic  Foot  is  a  cube,  each  of  whose 
aides  is  a  square  foot. 

3.  The  Cubic  Ton  is  chiefly  used  for  estimating  the  cartage  and 
transportation  of  timber.     By  a  ton  of  round  timber  is  meant,  such 
a  quantity  of  timber  in  its  rough  or  natural  state,  as  when  hewn, 
will  make  40  cubic  feet,  and  is  supposed  to  be  equal  in  weight  to 
50  feet  of  hewn  timber. 

4.  The  cubic  ton  or  load,  is  by  no  means  an  accurate  or  uniform 
standard  of  estimating  weight;   for.  different  kinds  of  timber,  are  of 
very  different  degrees  of  density.     But  it  is  perhaps  sufficiently  ac- 
curate'for  the  purposes  to  which  it  is  applied. 

QUEST. — 97.  In  what  is  Cubic  Measure  used?  Recite  the-  Table.  How 
lony,  wide,  and  high,  must  a  pile  of  wood  be  to  make  a  cord  V  What  is  a 
cube?  What  is  a  cubic  inch?  What  is  a  cubic  foot  V  C:m  you  draw  a  cubit 
Inch  on  your  slate  ? 

tJ1Tl  V  EIwi1 


94  COMPOUND  [SECT.  VII. 

WINE  MEASURE. 

98.  Wine  Measure  is  used  in  measuring  wine,  alco- 
hol, molasses,  oil,  and  all  other  liquids  except  boer,  ale, 
and  milk. 

4  gills  (gi.)  make  1  pint,         marked         pt. 

2  pints  "     1  quart,  "  qt. 

4  quarts  *     1  gallon,  "  gal 

3 1£  gallons  "     1  barrel,  "  bar.-o\'bbl. 

4*2  gallons  "     1  tierce,  "  tier. 

63  gallons,  or  2  bbls.  "     1  hogshead,       "  hhd. 

2  hogsheads  "     1  pipe  or  butt,  "  pi. 

2  pipes  "     1  tun,  "  tun. 

OBS.  The  wine  gallon  contains  231  cubic  inches. 

BEER  MEASURE. 

99.  Beer  Measure  is  used  in  measuring  beer,  ale,  and 
milk. 

2  pints  (pt.)          make  1  quart,     marked  qt. 

4  quarts  "     1  gallon,         "  gal. 

36  gallons  "     1  barrel,          "     lar.  or  bbl. 

54  gals,  or  H  bbls.  "     1  hogshead,    "  hhd. 

OBS.  The  beer  gallon  contains  282  cubic  inches.    In  many  places 

milk  is  measured  by  wine  measure. 

DRY  MEASURE. 

100.  Dry  Measure  is  used  in  measuring  grain,  fruit, 
silt,  &c. 

2  pints  (pts.)        make  1  quart,  marked      qt. 

8  quarts  "     1  peck,  "  pk. 

4  pecks,  or  32  qts.  "     1  bushel,  "  bu. 

8  bushels  "     1  quarter,  "  qr. 

32  bushels  "     1  chaldron,  "  ch. 

Note. — In  England,  36  bushels  of  coal  make  a  chaldron. 

QUEST.— 98.  In  what  is  Wine  Measure  used  ?  Recite  the  Table.  Obs.  How 
many  cubic  inches  in  a  wine  gallon?  99.  In  what  is  Beer  Measure  used? 
Repeat  the  Table.  Obs.  How  many  cubic  inches  in  a  beer  gallon  ? 


ARTS.  98 — 102.J  NUMBERS.  Jo 

TIME. 

1O1«  Time  is  naturally  divided  into  days  and  years  ; 
the  former  are  caused  by  the  revolution  of  the  Earth  on  its 
axis,  the  latter  by  its  revolution  round  the  Sun. 
60  seconds  (sec.)  make  1  minute,      marked  min. 

CO  minutes  "      1  hour,  "       hr. 

24  hours  "      1  day,  "       d. 

7  days  "      1  week,  "       ivk. 

4  weeks  "      1  lunar  month,  "       mo. 

1 2  calendar  months,  or     )    .       _    .  ., 

,   v  >   f       1  civil  year,  yr. 

3 65 -days,  6  hrs.,  (nearly,)  $ 

13  lunar  mo.,  or  52  weeks,    "      1  year,  "       yi\ 
100  years                                  "      1  century,           "       cm. 

OBS.  1.  Time  is  measured  by  clocks,  watched,  chronometers,  dials, 
hour-glasses,  <fcc. 

2.  A  civil  year  is  a  legal  or  common  year  ;  a  period  of  time  es 
tablished  by  government  for  civil  or  common  purposes. 

3.  A  solar  year  is  the  time  in  which  the  earth  revolves  round 
the  sun,  and  contains  365  days,  5  hours,  48  min.,  and  48  sec. 

4.  A  leap  year,  sometimes  called  bissextile,  contains  366  days, 
and  occurs  once  in  four  years. 

It  is  caused  by  the  excess  of  6  hours,  which  the  civil  year  con- 
tains above  365  days,  and  is  so  called  because  it  leaps  or  rims  over 
one  day  more  than  a  common  year.  The  odd  day  is  added  to  Feb- 
ruary, because  it  is  the  shortest  month.  Every  leap  year,  there- 
fore, February  has  29  days. 

1O2*  The  names  of  the  days  are  derived  from  the 
names  of  certain  Saxon  deities,  or  objects  of  worship.  Thus> 

Sunday  is  named  from  the  sun,  because  this  day  was  dedicated 
to  its  worship. 

Monday  is  named  from  the  moon,  to  wliich  it  was  dedicated. 

QVEST.— 100.  In  what  is  Dry  Measure  used  ?  Recite  the  table.  101.  Ho? 
is  Time  naturally  divided  ?  How  are  the  former  caused  ?  How  the  latter  \ 
ilepeat  the  Table.  Obs.  How  is  Time  measured  ?  What  is  a  civil  year  ?  A 
polar  year?  A  leap  year?  How  is  Leap  Year  caused  ?  To  which  month  ij 
the  odd  dav  added  ?  From  what  are  the  names  of  the  days  derived  ? 


96  COMPOUND  [SECT.  VIL 

Tuesday  is  derived  from  Tuiaco,  the  Saxou  god  of  war. 
Wednesday  is  derived  from  Woden,  a  deity  of  northern  Europe. 
Thursday  is  from  Thor,  the  Danish  god  of  thunder,  storms,  <fcc. 
Friday  is  from  Friga,  the  Saxon  goddess  of  beauty. 
Saturday  is  from  the  planet  Saturn,  to  which  it  was  dedicated 

1O3.  The  following  are  the  names  of  the  12  calendar 
months,  with  the  number  of  days  in  each  : 

month,  has  31  days. 

\      "         "    28     u 

"          u    31     u 
,       u         u    30     u 


31 
31 


u          *    30     u 
a         "    31     a 

OBS.  1.  The  oumber  of  days  in  each  month  may  be  easily  re 
B&suabered  from  the  following  lines  : 

"  Thirty  days  kith  September, 
April,  June,  and  November  ; 
February  twenty-eight  alone, 
All  the  rest  have  thirty-one  ; 
Except  in  Leap  Year,  then  is  the  time, 
When  February  has  twenty-nine." 

2.  The  names  of  the  calendar  mouths  were  borrowed  from  the 
Romans,  and  most  of  them  had  a  fanciful  origin.  Thus, 

January  was  named  after  Janus,  a  Roman  deity,  who  was  sup- 
posed to  preside  over  the  year,  and  the  commencement  of  all 
undertakings  . 

February  was  derived  fromfebrno,  a  Latin  word  which  signifies 
to  purify  by  sacrifice,  and  was  so  called  because  this  month  was 
devoted  to  the  purification  of  the  people, 


January, 

(Jan.)    the  first 

February, 

(Feb.)      "  secon 

March, 

(Mar,)     «  third 

April, 

(Apr.)     "  fourt 

May, 

(May)      "  fifth 

June, 

(June)     "  sixth 

July, 

(July)      "  seven 

August, 

(Aug.)     "  eighi 

September, 

(Sept.)     u  nintl 

October, 

(Oct.)      "  tenth 

November, 

(Nov.)     "  devei 

December, 

(Dec.)      «  twelj 

QUEST,— I.03L  Wliat  is  the  origin  of  iho  »ames  of  the  months  J 


ARTS.  103 — 105.]  NUMBERS.  9? 

March  was  named  after  Mars,  the  Roman  god  of  war ;  and  was 
originally  the  first  month  of  the  Roman  year. 

April,  from  the  Latin  aperio,  to  open,  was  so  called  from  th« 
opening  of  buds,  blossoms,  <fcc.,  at  this  season. 

May  was  named  after  the  goddess  Maia,  the  mother  of  Mercury, 
to  whom  the  ancients  used  to  offer  sacrifices  on  the^rs^  day  ct 
this  month. 

June  was  named  after  the  goddess  Juno,  the  wife  of  Jupiter. 

July  was  so  called  in  honor  of  Julius  Casar,  who  was  born  in 
this  month. 

August  was  so  called  in  honor  of  Augustus  Ceesar,  a  Roman 
Emperor,  who  entered  upon  his  first  consulate  in  this  month. 

September,  from  the  Latin  numeral  septem,  seven,  was  so  called, 
because  it  was  originally  the  seventh  month  of  the  Roman  year. 
It  is  the  ninth  month  in  our  year. 

October,  from  the  Latin  octo,  eight,  was  so  called  because  it  was 
the  eighth  month  of  the  Roman  year. 

November,  from  the  Latin  novem,  nine,  was  so"  called  because  it 
was  the  ninth  month  of  the  Roman  year. 

December,  from  the  Latin  decem,  ten,  was  so  called  because  it 
was  the  tenth  month  of  the  Roman  year.  •»' 

1O4»  The  year  is  also  divided  \ntofour  seasons  of 
three  months  each,  viz:  Spring,  Summer,  Autumn  or 
Fall,  and  Winter. 

Spring  comprises  March,  April,  and  May  ;  Summer, 
June,  July,  and  August ;  Autumn  oY  Fall,  September, 
October,  and  November ;  Winter,  December,  Jan.  and  Feb. 

CIRCULAR  MEASURE. 

1O5.  Circular  Measure  is  applied  to  the  divisions  of 
the  circle,  and  is  used  in  reckoning  latitude  and  longitude, 
and  the  motion  of  the  heavenly  bodies. 

60  seconds  (")         make  1  minute,      marked  ' 
60  minutes  "      1  degree,  "       ° 

30  degrees  "      1  sign,  "       s. 

12  signs,  or  360°         "      1  circle,  "       c. 

QUEST. — 104.  Name  the  seasons.    105.  To  what  is  Circular  Measure  applied  1 


98 


COMPOUND 


|  SECT.  VII. 


OBS.  1.  Circular  Measure  is 
often  called  Angular  Measure, 
and  is  chiefly  used  by  astrono- 
rrers.  navigators,  and  surveyors. 

2.  The  circumference  of  every 
c;rcle  is  divided,  or  supposed  to 
b^  divided,  into  360  equal  parts, 
called  degrees,   as  in  the  sub- 
joined figure. 

3.  Since  a  degree  is  -yeTf  part 
Oi  the  circumference  of  a  circle, 
it   is    obvious    that    its    length 
must  depend  on  the  size  of  the  circle 


MISCELLANEOUS  TABLE. 

1O6«  The  /oil owing  denominations  not  included    in 
**\e  preceding  Tables,  are  frequently  used. 

12  units 

12  dozen,  or  144 

12  gross,  or  1728 

20  units 

56  pounds 
100  pounds 

30  gallons 

200  Ibs.  of  shad  or  salmon 
196  pounds 
200  pounds 

14  pounds  of  iron,  or  lead 
21£  stone 
8  pigs 

OBS.  Formerly  112  pounds  were  allowed  for  a  quintal. 

GUEST. — Obs.  What  is  Circular  Measure  sometimes  called  7  By  whom  is  it 
chiefly  used  ?  Into  what  is  the  circumference  of  every  circle  divided  1  On 
what  does  the  length  of  a  degree  depend  ?  108.  How  many  units  make  a 
dozen  ?  How  many  dozen  a  gross  ?  A  great  gross  ?  How  many  units  mako 

;;  .-.c ..re  1    Pounds  a  Srkin  ? 


make  1  dozen,  (doz.) 
"  I 
"  1 
"  1 
"  1 
"  1 
"  1 

1 

1 

1 


great  gross, 
score. 

firkin  of  butter, 
quintal  of  fish, 
bar.  of  fish  in  Mass. 
bar.  in  N.  Y.  and  Ct. 
bar.  of  flour, 
bar.  of  pork. 
1  stone. 

1  P^ 
1  fother. 


ARTS.  106 — 108.]        NUMBERS.  99 

PAPER  AND  BOOKS. 

IO7»  The  terms,  folio,  quarto,  octavo,  <fec.,  applied  to 
books,  denote  the  number  of  leaves  into  which  a  sheet 
of  paper  is  folded. 

24  sheets  of  paper  make  1  quire. 

20  quires  "  1  ream. 

2  reams  "  1  bundle. 

5  bundles  "  1  bale. 

A  sheet  folded  in  two  leaves,  is  called  &  folio. 
A  sheet  folded  in  four  leaves,  is  called  a  quarto,  or  4  to. 
A  sheet  folded  in  eight  leaves,  is  called  an  octavo,  or  Svo. 
A  sheet  folded  in  twelve  leaves,  is  called  a  duodecimo. 
A  sheet  folded  in  sixteen  leaves,  is  called  a  16 wo. 
A  sheet  folded  in  eighteen  leaves,  is  called  an  I8?no. 
A  sheet  folded  in  thirty-two  leaves,  is  called  a  3  2 wo. 
A  sheet  folded  in  thirty-six  leaves,  is  called  a  3 6 wo. 
A  sheet  folded  in  forty-eight  leaves,  is  called  a  48wo. 

1O8.  Previous  to  the  adoption  of  Federal  money  in 
1786,  accounts  in  the  United  States  were  kept  in  pounds 
shillings,  pence,  and  farthings. 

In  New  England  currency,  Virginia,  Ken-  i 

tucky,  Tennessee.  Indiana,  Illinois,  Mis-  >6  shil.  make  $1. 

souri,  and  Mississippi,  3 

In  New  York  currency,  North  Carolina,  i 

Ohio,  and  Michigan,  {  8  shil  make  &'1 • 

In    Pennsylvania   currency,   New   Jersey,  i 

Delaware,  and  Maryland,  "    \  ls'  Gd*  make  $1" 

In  Georgia  currency  and  South  Carolina.       4s.  8d.  make  $1. 
In  Canada  currency,  and  Nova  Scotia,  5  shil.  make  $1. 

(it'KST. — 107.  When  a  sheet  of  paper  is  folded  in  two  leaves,  what  is  it 
called  ?  When  in  four  leaves,  what  ?  When  in  eight  1  In  twelve  ?  In 
sixteen  ?  In  eighteen  ?  In  thirty-six  ?  108.  Previous  to  the  adoption  of  Fed- 
eral Money,  in  what  were  accounts  kept  in  the  U.  S.  1  How  many  shillings 
make  a  dollar  in  N.  E.  currency?  In  N.  Y.  currency?  In  Penn.  currency? 
ij  f'rc.'gia  currency  ?  Tn  Canada  e'.imMsoy  ? 


100  COMPOUND  [SECT.  VII. 

OBS.  At  the  time  Federal  money  was  adopted,  the  colonial  cur- 
rency or  bills  of  credit  issued  by  the  colonies,  had  more  or  less  de- 
preciated in  value :  that  is,  a  Colonial  pound  was  worth  less  than  a 
pound  Sterling ;  a  colonial  shilling,  than  a  shilling  Sterling.  &c. 
This  depreciation  being  greater  in  some  colonies  than  in  oth.trs 
gave  rise  to  the  different  values  of  the  State  currencies. 

ALIQUOT    PARTS    OF    $1    IN    FEDERAL    MONEY. 


50  cents  =  $£ 
33i  cents  =  $£ 
25  cents  =,  $i 
20  cents  =  $£ 


cents  = 


12|  cents 

10     cents 

8i  cents 

6|  cents 

5     cents 


PARTS  OF  $  1  IN  NEW  YORK  CURRENCY. 


4  shillings  = 

2  shil.  8  pence      = 


1  shil.  4  pence   =  $£ 
1  shilling  =  $i 

6  pence  =  $-fV 


2  shillings 

OBS.  1.  In  New  York  currency,  it  will  be  aeen,  (Art.  108,)  that 
A  six-pence,  written    6d.  =     6£  cents. 

A  shilling,  "          Is.  =  12£      " 

One  (shil.)  and  6  pence,         "       1/6.  =  18|      " 
Two  shillings,  "          2s.  =  25        " 

PARTS  OF  $1  IN  NEW  ENGLAND  CURRENCY. 


3  shillings  = 

2  shillings  = 

1  shil.  and  6d.      = 


1  shilling 
9  pence 
6  pence 


OBS.  2.  In  New  England  currency,  it  will  be  seen,  that 

A  four-pence-half-penny,  written  4£d.  =     6$  cents. 
A  six-pence,                             "          6d.  =     8§      " 
A  nine-pence,                          "          9d.  =  12^      " 
A  shilling,                                  «  Is.  =  16§      '.« 

One  (shil.)  and  six-pence,      «  1/6.  =  25        " 
Two  shillings,                          "          2s.  =  33*      " 

QUEST.— What  are  the  aliquot  parts  of  $1  in  Federal  Money  ?  In  New  York 
currency  1  In  New  England  currency  1  What  are  the  aliquot  parts  of  a  pound 
Sterling  7  Of  a  shilling  1 


ART.  108.] 


NUMBERS. 


101 


ALIQUOT    PARTS 

Parts  of  £1. 

10  shil.  =  ££ 

6s.  8d.  =  ££ 

5  shil.  =  £i 

4  shil.  =  ££ 

3s.  4d.  =  ££ 

2s.  6d.  =  £i 

2  shil.  = 

Is.  8d.  = 

1  shil.  = 


OF    STERLING    MONEY. 

Parts  of  Is. 
6  pence  =  £  shil. 
4  pence  =  -J-  shil. 
3  pence  =  -^  shil. 
2  pence  =  -J-  shil. 
1-J-  pence  =  i  shil. 
1  penny  =  -jV  shil. 

1  far.  =  -J-  penny. 

2  far.  =  £  penny. 

3  far.  =  f  penny. 


ALIQUOT    PARTS    OF    A    TON. 


10  hund.  lbs.=£  ton. 
5  hund.  lbs.=i  ton. 
4  hund.  lbs.=£  ton. 


2  hund.  2  qrs.=i  ton. 
2  hund.  Ibs.  =1^  ton. 
1  hund.  Ibs.  =•£$  ton. 


ALIQUOT    PARTS    OF    A    POUND    AVOIRDUPOIS. 


8  ounces =-J-  pound. 
4  ounces =-J-  pound. 

ALIQUOT 

[  year. 
-\  year. 
4  year. 
-\  year. 
=-J-  year. 
-\  year, 
-i  year. 
=-iV  year. 


Parts  of  1 
6    months 
4    months 
3    months 
2    months 
1|  month 
1-|  month 
1    month 


2  ounces  =^  pound. 
1  ounce  =-fV  pound. 

PARTS    OF    TIME. 

Parts  of  I  month. 
15  days=i-     month. 
10  days=-£     month. 

6  days=-§-     month. 

5  days=-J-     month. 

3  days=tV  month. 

2  days=iV  month. 

1  day  =^V  month. 


QUEST.—  How  many  shillings  in  half  a  pound  Ster.  ?  In  a  fourth  ?  A  fifth  ? 
A  tenth  ?  A  twentieth  ?  How  many  pence  in  half  a  shilling  ?  In  a  third  ?  A 
fourth  ?  A  sixth  ?  A  twelfth  ?  How  many  hundreds  in  half  a  ton  ?  In  a 
fourth  ?  A  fifth  ?  A  tenth  ?  How  many  ounces  in  half  a  pound  ?  In  a  fourth  t 
An  eighth  ?  A  sixteenth  ?  How  many  months  in  half  a  year  ?  In  a  third  ? 
A  fourth?  A  sixth?  A  twelfth? 


102  FEDERAL  MONEY.  [SECT.  VIII. 

SECTION    VIII. 

FEDERAL   MONEY. 

1  1 0.  Accounts  in  the  United  States  are  kept  in  dol- 
lars, cents,  and  mills.  Eagles  are  expressed  in  dollars,  and 
dimes  in  cents.  Thus,  instead  of  4  eagles,  we  say,  40  dol- 
lars ;  instead  of  5  dimes,  we  say,  50  cents,  &c. 

111.  Dollars  are  separated  from  cents  by  placing  a 
point  or  separatrix  (  .  )  between  them.     Hence, 

112.  To  read  Federal  Money. 

Call  all  the  figures  on  the  left  of  the  point,  dollars  ;  the 
first  two  figures  on  the  right  of  the  point,  are  cents  ;  the 
third  figure  denotes  mills  ;  the  other  places  on  the  right  are 
parts  of  a  mill  Thus,  $5.2523  is  read,  5  dollars,  25 
cents,  2  mills,  and  3  tenths  of  a  mill. 

OBS.  1.  Since  two  places  are  assigned  to  cents,  when  no  cents 
are  mentioned  in  the  given  number,  two  ciphers  must  be  placed 
before  the  mills.  Thus,  5  dollars  and  7  mills  are  written  $  5.007. 

2.  If  the  given  number  of  cents  is  less  than,  ten,  a  cipher  must 
always  be  written  before  them.  Thus,  8  cents  are  written  .08,  &c. 

1.  Read  the  following  expressions :  $83.635;  $75.50. 
$126.607;  $268.05;  $382.005;  $2160. 

2.  Write  the  following  sums :  Sixty  dollars  and  fifty 
cents.     Seventy-five  dollars,  eight  cents,  and  three  mills. 
Forty-eight  dollars    and   seven    mills.     Nine   cents.     Six 
cents  and  four  mills. 


QUKST. — 88.  What  is  Federal  Money?  What  are  its  denominations?  Re- 
cite the  Table.  110.  How  are  accounts  kept  in  the  United  States?  How  «ro 
Eagles  expressed?  Dimes?  111.  How  are  dollars  distinguished  from  cents 
and  mills?  112.  How  do  you  read  Federal  Money?  Obs.  How  many  places 
are  assigned  to  cents?  When  the  number  of  cents  is  less  than  ten,  what  must 
bo  done  ?  When  no  cents  are  mentioned,  what  do  you  do  ? 


ARTS.  110 — 113.]      FEDERAL  MONEY.  103 

REDUCTION   OF   FEDEKAL  MONEY. 

CASE    I. 
Ex.  1.  How  many  cents  are  there  in  65  dollars  ? 

Suggestion. — Since  in  1  dollar  there  are      Operation. 
100  cents,  in  05  dollars  there  are  65  times  as     65 
many.     Now,  to  multiply  by  10,  100,  &c.,  we       1QQ 
annex  as  many  ciphers  to  the  multiplicand,     65°0  cents. 
as  there  are  ciphers  in  the  multiplier.     (Art.  45.)     Hence, 

1 13.  To  reduce  dollars  to  cents,  annex  TWO  ciphers. 
To  reduce  dollars  to  mills,  annex  THREE  ciphers. 
To  reduce  cents  to  mills,  annex  ONE  cipher. 

OBS.  To  reduce  dollars  and  cents  to  cents,  erase  the  siyn  of  dollars 
and  the  separatrix.  Thus,  $25.36  reduced  to  cents,  become  2536 
cents. 

2.  Reduce  $4  to  cents.  Ans.  400  cents. 

3.  Reduce  $15  to  cents.  7.  Reduce  $96  to  mills. 

4.  Reduce  $27  to  cents.  8.  Reduce  $12.23  to  cents. 

5.  Reduce  $85  to  cents.  9.  Reduce  $86.86  to  cents. 

6.  Reduce  $93  to  cents.  10.  Reduce  $9.437  to  mills. 

CASE    II. 
1.  In  2345  cents,  how  many  dollars? 

Suggestion. — Since  100  cents  make  1  dol-  Operation, 
lar,  2345  cents,  will  make  as  many  dollars  l|00)23|45 
as  100  is  contained  times  in  2345.  Now  to  Ans.  $23.45 
divide  by  10,  100,  <fcc.,  we  cut  off  as  many 
figures  from  the  right  of  the  dividend  as  there  are  ciphers 
in  the  divisor.  (Art.  67.)  Hence, 


QUEST. — 113.  How  are  dollars  reduced  to  cents?    Dollars  to  mills?    Cents 
to  mills  ?     Obs.  Dollars  and  cents  to  cents  ? 


104  FEDERAL  MONEI  [SECT.  V1IL 

114*  To  reduce  cents  to  dollars. 

Point  off  TWO  figures  on  the  right  ;  the  figures  remain- 
ing  on  the  left  express  dollars  ;  the  two  pointed  off,  cents. 

115*  To  reduce  mills  to  dollars. 

Point  off  THREE  figures  on  the  right  ;  the  remaining 
figures  express  dollars  ;  the  first  two  on  the  right  of  the 
point,  cents  ;  the  third  one,  mills. 

116*  To  reduce  mills  to  cents. 

Point  off  ONE  figure  on  the  right,  and  the  remaining 
figures  express  cents  ;  the  one  pointed  off,  mills. 

2.  Reduce  236  cts.  to  dolls.  3.  Reduce  21  63  cts.  to  dolls. 
4.  Reduce  865  mills  to  dolls.  5.  Reduce  906  mills  to  cts. 
6.  Reduce  2652  cts.  to  dolls.  7.  Reduce  3068  cts.  to  dolls. 

ADDITION  OF  FEDERAL  MONEY. 

Ex.  1.  What  is  the  sum  of  $8.125,  $12.67,  $3.098,  $11  ? 


Suggestion.—  Write  the  dollars   under 

dollars,    cents   under   cents,    mills   under  12*67 

mills,  and  proceed  as  in  Simple  Addition.  3  098 

From  the  right  of  the  amount  point  off  11  .00 

three  figures  for  cents  and  mills.  ^nSt  $34.393 


1  17»  Hence,  we  derive  the  following  general 

RULE   FOR   ADDING   FEDERAL   MONEY. 
Write  dollars  under  dollars,  cents  under  cents,  mills 
under  mills,  and  add  each  column,  as  in  simple  numbers. 
From  the  right  of  the  amount,  point  off  as  many  figures 
for  cents  and  mills,  as  there  are  places  of  cents  and  mills 
in  either  of  the  given  numbers. 

QUEST.—  114.  How  are  cents  reduced  to  dollars?   115.  Mills  to  dollars?  117. 
How  do  you  add  Federal  Money?    How  point  off  the  amount? 


ARTS.  114 — 117.J      FEDERAL  MONEY.  105 

OBS.  If  either  of  the  given  numbers  have  no  cents  expressed, 
supply  their  place  by  ciphers. 

(2.)                    (3.)                    (4.)  (5.) 

$375.037         $4869.45         $760.275  $4607.375 

60.20                344.00            897.008  896.084 

843.462           6048.07            965.054  95.873 


(6.) 

$782.206 

$609.352 

(8.) 
$2903.76 

(9.) 

$4668.253 

84.60 

830.206 

453.06 

430.064 

379.007 

408.07 

25.89 

307.60 

498.015 

631.107 

6842.07 

7452.349 

10.  What  is  the  sura  of  $63.072,  $843.625,  and  $71.60  ? 

11.  Add  $873.035,  $386.23,  $608.938,  $169.176. 

12.  Add  463  dolls.  7  cts.;  248  dolls.  15  cts.;  169  dolls. 
9  cts.  7  mills. 

13.  Add  89  dolls.  8  ets.;  97  dolls.  10  cts.  3  mills;  40 
dolls.  6  cts.;  75  dolls. 

14.  Add  365  dolls.  20  cts.  2  mills;   68  dolls.  6  cts.  5 
mills  ;  7  cts.  3  mills ;  286  dolls. ;  80  dolls.  6  mills ;  30  dolls. 
15  cts. 

15.  A  man  bought  a  cow  for  $16.375,  a  calf  for  $4.875, 
and  a  ton  of  hay  for  $13.50:  how  much  did  he  pay  for 
the  whole  ? 

]  6.  A  lady  paid  $23  for  a  cloak,  $7.625  for  a  hat,  $25.75 
for  a  muff,  and  $18  for  a  tippet:  how  much  did  she  pay 
for  all  ? 

17.  A  farmer  sold  a  cow  for  $16.80,  a  calf  for  $4.08,  a 
horse  for  $78,  and  a  yoke  of  oxen  for  $63.18  :  how  much 
did  he  receive  for  all  ? 


QUEST.—  Obit.  When  any  of  the  given  numbers  have  no  cents  expressed 
how  is  their  place  supplied? 


106  FEDERAL    MONEY.  [SECT.  VIII. 

SUBTRACTION  OF  FEDERAL  MONEY. 

Ex.  1.  What  is  the  difference  between  $845.634,  and 
$86.087  ? 

Suggestion. — Write  the  less  number          Operation. 
under  the  greater,  dollars  under  dollars,  $845.634 

&c.,  then  subtract,  and  point  off  the  an-  86.087 

swer  as  in  addition  of  Federal  Money.          Ans.  $759.547- 

118.  Hence,  we  derive  the  following  general 

RULE   FOR   SUBTRACTING   FEDERAL   MONEY. 
Write  the  less  number  under  the  greater,  with  dollars 
under  dollars,  cents  under  cents,  and  mills  under  mills  ; 
the)i  subtract,  and  point   off  the  answer  as  in  addition 
of  Federal  Money. 


(2.) 
Prom  $856.453 
Take  $387.602 

(3.) 

$960.78 
$463.05 

(4-) 

$605.607 
78.36 

(5.) 
$6243.760 
327.053 

(6.) 
From  965.005 
Take   87.85 

(*•) 

840.000 

378.457 

(8.) 
483.853 
48.75 

(9.) 
4265.76 
2803.98 

10.  From  $86256.63  take  $4275.875  ? 

11.  From  $100250,  take  $32578.867  ? 

12.  From  1  dollar,  subtract  11  cents. 

13.  From  3  dolls.  6  cts.  7  mills,  take  75  cents. 

14.  From  110  dolls.  8  mills,  take  60  dolls,  and  8  cents. 

15.  From  607  dolls.  7  cents,  take  250  dolls,  and  3  cts. 

16.  A  lad  bought  a  cap  for  $2.875,  and  paid  a  five- 
dull  ar-bill  :  how  much  change  ought  he  to  receive  back  ? 

17.  Henry  has  $7.68,  and  William  lias  $9.625  :  how 
much  more  has  the  latter  than  the  former  ? 

18.  From  $865275.60,  take  $340076.875. 


QUEST.— 10G.  How  do  you  subtract  Federal  Money  ?    How  point  off  the 
answer  ? 


ARTS.  118, 119.]        FEDERAL  MONEY.  107 

MULTIPLICATION  OF  FEDERAL  MONEY. 

Ex.  1.  What  will  3  caps  cost,  at  $1.625  apiece  ? 

Suggestion.— Since  1  cap  costs  $1.625,  Operation. 
3  caps  will  cost  3  times  as  much.  We  Si  625 

therefore  multiply  the  price  of  1  cap  by  3,  3 

the   number  of  caps,  and  point  off  three     Ans.  $4.375 
places  for  cents  and  mills.     Hence, 

119.  When /the  multiplier  is  a  whole  number,  we  have 
the  following 

RULE   FOR   MULTIPLYING   FEDERAL   MONEY. 

Multiply  as  in  simple  numbers,  and  from  the  right  of 
the  product,  point  off  as  many  figures  for  cents  and  mills, 
as  there  are  places  of  cents  and  mills  in  the  multiplicand. 

OBS.  1.  In  Multiplication  of  Federal  Money,  as  well  as  in  simple 
numbers,  the  multiplier  must  always  be  considered  an  abstract 
number. 

2.  In  business  operations,  when  the  mills  are  6  or  over,  it  is 
customary  to  call  them  a  cent ;  when  under  5,  they  are  disregarded. 

(2.)  (3.)  (4.)  (5.) 

Multiply    $633.75       $805.625       $879.075       $9071.26 
By  8  9  24  37 


(6.) 

(7.) 

(8.) 

(9.) 

Multiply  $ 

4063.36 

$5327.007 

$6286.69 

$8265.68 

By 

63 

86 

123 

264 

10.  What  cost  8  melons,  at  17  cents  apiece  ? 

11.  What  cost  12  lambs,  at  87  cents  apiece  ? 

12.  What  cost  8  hats,  at  $3.875  apiece  ? 

13.  At  $8.75  a  yard,  what  will  9  yards  of  silk  come  tot 

14.  At  $1.125  apiece,  what  will  11  turkeys  cost? 

QUEST.— 119.  How  do  you  multiply  Federal  Money  ?  How  point  off  the 
product  ?  Obs.  What  must  tho  multiplier  always  be  considered  ?  When 
th«  mills  are  5,  or  over,  what  is  it  customary  tt  call  them.  ?  When  less  than 
5,  what  may  be  done  with  them  * 


108  FEDERAL    MONET.  [SECT.  VITT. 

15.  At  $2.63  apiece,  what  will  15  chairs  come  to  ? 
1(3.  What  costs  25  Arithmetics,  at  37£  cents  apiece? 

17.  What  cost  38  Readers,  at  62^-  cents  apiece? 

18.  What  cost  46  over-coats,  at  $25.68  apiece  ? 

19.  What  cost  69  oxen,  at  $48.50  a  head  ? 

20.  At  $23  per  acre,  what  cost  65  acres  of  land  ? 

21.  At  $75.68  apiece,  what  will  56  horses  come  to  ? 

22.  At  7-J  cents  a  mile,  what  will  it  cost  to  ride  100 
miles  ? 

23.  A  farmer  sold  84  bushels  of  apples,  at  87-J-  cents  per 
bushel :  what  did  they  come  to  ? 

24.  If  I  pay  $5.3 7£  per  week  for  board,  how  much  will 
it  cost  to  board  52  weeks  ? 

DIVISION  OF  FEDERAL  MONEY. 
Ex.  1.  If  you  paid  $18.876  for  3  barrels  of  flour,  how 
much  was  that  a  barrel  ? 

Suggestion. — Since  3  barrels  cost  $18.-     n^^f^ 

t  \7UfilCiliiO7lit 

876,  1  barrel  will  cost   1   third  as  much.       g\  $IQ  876 
We  therefore  divide  as  in  simple  division,       .        ^ 
and  point  off  three  places  for  cents  and 
mills,  because  there  are  three  in  the  dividend.     Hence, 

1  2O.  When  the  divisor  is  a  whole  number,  we  have 
the  following 

RULE   FOR   DIVIDING   FEDERAL   MONEY. 
Divide  as  in  simple  numbers,  and  from  the  right  of  the 
quotient,  point  off  as  many  figures  for  cents  and  IT  ills,  as 
there  are  places  of  cents  and  mills  in  the  dividend. 

OBS.  When  the  dividend  contains  no  cents  and  mills,  if  there 
is  a  remainder  annex  three  ciphers  to  it ,  then  divide  as  before, 
and  point  off  three  figures  in  the  quotient. 

QUEST. — 120.  How  do  you  divide  Federal  Money  ?  How  point  off  the 
quotient  1  Obs.  When  the  dividend  contains  no  cents  and  mills,  how  proceed  ? 


120.1  FEDERAL    MONEY.  109 

Note. — For  a  more  complete  development  of  multiplication  and 
division  of  Federal  Money,  the  learner  is  referred  to  the  author's 
Practical  and  Higher  Arithmetics. 

When  the  multiplier  or  divisor  contain  decimals,  or  cents  and 
mills,  to  understand  the  operation  fully,  requires  a  thorough 
knowledge  of  Decimal  Fractions,  a  subject  which  the  limits  of  this 
work  will  not  allow  us  to  introduce. 

(2.)  (3.)  (4.) 

6)  $856.272.  8)  $9567.648.  9)  $7254.108. 

5.  Divide  $868.36  by  27.       6.  Divide  $3674.65  by  38. 

7.  Divide  $486745  by  49.      8.  Divide  $634.07,5  by  56. 

9.  Divide  $6634.25  by  60.  10.  Divide  $5340.73  by  78. 
11.  Divide  $7643.85  by  83.  12.  Divide  $4389.75  by  89. 
13.  Divide  $836847  by  94.  14.  Divide  $94321.62  by  97. 

15.  A  man  paid  $2563.84  for  63  sofas  :  what  was  that 
apiece  ? 

16.  A  miller  sold  86  barrels  of  flour  for  $526.50  :  how 
much  was  that  per  ban-el  ? 

17.  If  a  man  pays  $475.56  for  65  barrels  of  pork,  what 
is  that  per  barrel  ? 

18.  A  man  paid  $1875.68  for  93  stoves  :  how  much 
was  that  apiece  ? 

19.  If  $2682.56  are  equally  divided   among  100  men, 
how  much  will  each  receive  ? 

20.  A  cabinet-maker  sold  116  tables  for  $968.75  :  how 
much  did  he  get  apiece  ? 

21.  A  farmer  sold  168  sheep  for  $465  :  how  much  did 
he  receive  apiece  for  them  ? 

22  A  miller  bought  216  bushels  of  wheat  for  $375.50  : 
how  much  did  he  pay  per  bushel  ? 

23.  If  $2368.875  were  equally  divided  among  348  per 
sons, how  much  would  each  person  receive? 


110  REDUCTION.  [SECT.  IX. 

SECTION    IX. 

REDUCTION. 

ART.  121.  REDUCTION  is  the  process  of  changing 
Compound  Numbers  from  one  denomination  into  another, 
without  altering  their  value. 

REDUCING  HIGHER  DENOMINATIONS  TO  LOWER. 

122.   Ex.  1.  Reduce  £2,  to  farthings. 

Suggestion.  —  First    reduce    the  Operation 

given    pounds   (2)    to    shillings,   by  £2 

multiplying   them    by   20,   because  20s.  in  £l. 

20s.   make   £l.     Next   reduce  the  40  shillings, 

shillings  (40)  to   pence,  by  multi-  12d.  in  Is. 

plying   them  by   12,  because   12d.  480  pence. 

make  Is.     Reduce  the  pence  (480)  !far- in  ld- 

to   farthings,  by  multiplying  them     ^ns.  1920  farthings, 
by  4,  because  4  far.  make  Id. 

2.  Reduce  £l,  2s.  4d.  and  3  far.  to  farthings. 

Suggestion.  —  In     this     example  Operation. 

there  are  shillings,  pence,  and  far-  £    s>  d.  far. 

things.     Hence,  when   the    pounds 

20s.  in  £1. 
are   reduced   to  shillings,  the  given 

shillings  (2)  must  be  added  men-  ^2  shillings. 

tally  to  J;he  product.     When   the 
,  .„.  i       ,  ,  268  pence, 

shillings  are  reduced  to  pence,  the  4  f •  •  '     11 

given   ponce   (4)   must  be  added ;     Ant^^^ 

and  when  the  pence  are  reduced  to 

farthings,  the  given  farthings  (3)  must  be  added. 

CIUEST.— 121.  What  is  reduction  ?  122.  Ex.  L  How  reduce  pounds  to  shil 
lings?  Why  multiply  by  20  ?  How  are  shillings  reduced  to  pence?  Why1 
Huw  pence  to  farthings  ?  Whj  ? 


ARTS.  121  —  124.]          REDUCTION.  Ill 

OBS.  In  these  examples  it  is  required  to  reduce  higher  denomi- 
nations to  lower,  as  pounds  to  shillings,  shillings  to  pence,  <fcc. 

1  253.  The  process  of  reducing  higher  denominations 
to  lower  i  is  usually  called  Reduction  Descending. 

It  consists  in  successive  multiplications,  and  may  with 
propriety  be  called  Reduction  by  Multiplication. 


From  the  preceding  illustrations  we  derive  the 
following 

RULE  FOR  REDUCTION  DESCENDING. 

Multiply  the  highest  denomination  given  by  the  num- 
ber required  of  the  next  lower  denomination  to  make  ONE 
of  this  higher,  and  to  the  product,  add  the  given  number  of 
this  lower  denomination. 

Proceed  in  this  manner  with  each  successive  denomina- 
tion, till  you  come  to  the  one  required. 

EXAMPLES. 

3.  Reduce  4  miles,  2  fur.,  8  rods  and  4  feet  to  feet. 

Operation. 

Suggestion.  —  Having  reduced  the  m-  fur-  r-  ft- 

miles  and  furlongs  to  rods,  we  have 
1368  rods.     We  then  multiply  by  — 

16£,  because  16-j-  feet  make  1  rod.  .« 

(Art.  94.)     Now    10*  is   a   mixed  2)^  rods. 

number;  we   therefore   first    multi-  16  1 

pty    Dy    the    whole    number    (16),  82V2~ 

then  by  the  fraction  (£),  and  add  1308 

the  products  together.     (Art.  84.)  684 

Am.  22576  feet. 


QUEST. — 123.  What  is  reducing  compound  numbers  to  lower  denominations 
usually  called?    Which  of  the  fundamental  rules  is  employed  in  reduction 

•ling?     124.  VVii.it'?-  . -liur.  I)oscc..vi;iji< '• 


112  REDUCTION.  [SECT.    IX. 

4.  In  £5, 16s.  7d.,  how  many  farthings  ?  Am.  5596  far. 

5.  In  £18  how  many  pence? 

6.  In  £23,  9s.,  how  many  shillings  ? 

7.  In  17s.  2d.  3  far.,  how  many  farthings? 

8.  Reduce  5  Ibs.  6  oz.  Troy  weight,  to  grains. 

Ans.  31680  grs. 

9.  Reduce  13  Ibs.  Troy,  to  ounces. 

10.  Reduce  4  Ibs.  3  oz.  Troy,  to  penny  weights. 

11.  Reduce  15  oz.  6  pwts.  4  grs.  Troy,  to  grains. 

12.  In  2  cwt.  3  qrs.  7  Ibs.  4  oz.  3  drams,  avoirdupois 
weight,  how  many  drams  ?     Ans.  72259  drams. 

13.  In  13  Ibs.  4  oz.  avoirdupois,  how  many  ounces? 

14.  In  2  qrs.  17  Ibs.  avoirdupois,  how  many  pounds? 

15.  In  6  Ibs.  12  oz.  avoirdupois,  how  many  drams? 

16.  In  12  cwt.  1  qr.  6  Ibs,  avoir.,  how  many  ounces? 

17.  In  16  miles,  how  many  rods? 

18.  In  28  rods  and  2  feet,  how  many  inches  ? 

19.  In  19  fur.  4  rods  and  2  yds.,  how  many  feet? 

20.  In  25  leagues  and  2  m.,  how  many  rods  ? 

21.  Reduce  14  yards  cloth  measure  to  quarters. 

22.  Reduce  21  yards  2  quarters  to  nails. 

23.  Reduce  17  yards  3  quarters  2  nails,  to  nails. 

24.  How  many  quarts  in  23  gallons,  wine  measure  ? 

25.  How  many  gills  in  30  gallons  and  2  quarts  ? 

26.  How  many  gills  in  63  gallons  ? 

27.  How  many  quarts  in  41  hogsheads? 

28.  How  many  pecks  in  45  bushels  ? 

29.  How  many  pints  in  3  pecks  and  2  quarts  ? 

30.  How  many  quarts  in  52  bu.  and  2  pecks  ? 

31.  How  many  hours  in  15  weeks? 

32.  How  many  minutes  in  25  days  ? 

33.  How  many  seconds  in  265  hours  ? 

34.  How  many  minutes  in  52  weeks? 

35.  How  many  seconds  in  68  days? 


ARTS.  125,  126.]  REDUCTION.  113 

KEDUCING  LOWER  DENOMINATIONS  TO  HIGHER.* 

125.  Ex.  1.  Reduce  1920  farthings  to  pounds. 
Suggestion. — First  reduce  the  given  far-     Operation. 

tilings  (1920)   to  pence,  the   next  higher        4)1920  fart 
denomination,  by  dividing  them  by  4,  be-        12)480d. 
cause   4   far.  make  Id.     Next  reduce   the          20)40s. 
pence  (480)  to  shillings,  by  dividing  them  £2  Ans. 

by  12,  because  12d.  make  Is.     Finally,  re- 
duce the  shillings  (40)  to  pounds,  by  dividing  them  by  20, 
because  20s.  make  £l.     The  answer  is  £2.     That  is,  1920 
far.  are  equal  to  £2. 

2.  In  1075  farthings,  how  many  pounds? 

Suggestion. — In    dividing   the  Operation. 

given  farthings   by  4,  there  is  a         4)1075  far. 
remainder  of  3  far.,  which  should          12)268d.  3  far.  over, 
be   pjaced   on  the  right.     In  di-  2 0)2 2s.  4d.  over, 

viding   the   pence  (268)   by    12,  £l,  2s.  over, 

there  is  a  remainder  of  4d.,  which  Ans.  £l,  2s.  4d.  3  far. 
should  also  be  placed  on  the 

right.  In  dividing  the  shillings  (22)  by  20,  the  quotient 
is  £1  and  2s.  over.  The  last  quotient  with  the  several 
remainders  is  the  answer.  That  is,  1075  far.  are  equal  to 
£1,  2s.  4d.  3  far. 

OBS.  In  the  last  two  examples,  it  is  required  to  reduce  lower  de- 
nominations to  higher,  as  farthings  to  pence,  pence  to  shillings,  <fec. 
The  operation  is  exactly  the  reverse  of  that  in  Reduction  Descending. 

126.  The  process  of  reducing  lower  denominations  to 
higher  is  usually  called  Reduction  Ascending. 

It  consists  in  successive  divisions,  and  may  with  propri- 
ety be  called  Reduction  by  Division. 

QUKST.— 125.  Ex.  1.  How  are  farthing*  reduced  to  pence  ?  Why  divide  by  4  ? 
How  reduce  p*nco  to  shillings  ?  Why?  How  shillings  to  pounds?  Why? 
126.  What  is  reducing  cojnpound  numbers  to  higher  denominations  usually 
called  1  Which  of  the  fundamental  rules  is  employed  in  Reduction  Ascending  1 


114  REDUCTION.  (SECT.  IX. 

*127.  From  the  preceding  illustrations  we  derive  the 
following 

RULE   FOR  REDUCTION   ASCENDING. 

Divide  the  given  denomination  by  that  number  which  it 
takes  of  this  denomination  to  make  ONE  of  the  next  higher. 
Proceed  in  this  manner  with  each  successive  denomination, 
till  you  come  to  the  one  required.  The  last  quotient,  with 
the  several  remainders,  will  be  the  answer  sought. 

128.  PROOF.  —  Reverse  the  operation  ;  that  is,  reduce 

bo.ck  the  answer  to  the  original  denominations,  and  if  the 

result  correspond  with  the  numbers  given,  the  work  is  right. 

OBS.  Each  remainder  is  of  the  same  denomination  as  the  divi- 

dend from  which  it  arose.  (Art.  51,  Obs.  2.) 

EXAMPLES. 

3.  In  429  feet,  how  many  rods  ?  Operation. 

Suggestion.  —  We  first  reduce  the  feet  3  )429  feet. 

to  yards,  then  reduce  the  yards  to  rods  5^)143  yds. 

by   dividing   them   by  5£.      (Art.  86.)  2         2 

Or,  we  may  divide  the  given  feet  by  11  )286 

16f  ,  the  number  of  feet  in  a  rod,  and  the  Ans.  26  rods 
quotient  will  be  the  answer. 

Proof. 

We  first  reduce  the  rods  back  to  yards*  26  rods. 

(Art.  84,)  then  reduce  the  yards  to  feet.  5^ 

The  result  is  429  feet,  which  is  the  same  130 

as  the  given  number  of  feet.  1  3 


Or,  we  may  multiply  the  26  by  16£,        143  yds. 
and  the  product  will  be  429.  3 

429  feet. 

4.  Reduce  256  pence  to  pounds.     Ans.  £l,  Is.  4d. 

5.  Reduce  324  pence  to  shillings. 

QUEST.  —  127.  What  is  the  rule  for  reduction  ascending?     128.  Ho\v  is  re 
ductJou  proved  ?     Ob*.  Of  what  denomination  is  each  remainder  ? 


ARTS.  127  — 129.]     REDUCTION.  115 

6.  Reduce  960  farthings  to  shillings. 

7.  Reduce  1250  farthings  to  pounds. 

8.  In  265  ounces  Troy  weight,  how  many  pounds? 

9.  In  728  pwts.,  how  many  pounds  Troy? 

10.  In  548  grains,  how  many  ounces  Troy? 

11.  In  638  oz.  avoirdupois  weight,  how  many  pounds? 

12.  In  736  Ibs.  avoirdupois,  how  many  quarters? 

13.  In  1675  oz.  avoirdupois,  how  many  hundred  weight  ? 

14.  In  1000  drams  avoirdupois,  how  many  pounds? 

15.  In  4000  Ibs.  avoirdupois,  how  many  tons? 

16.  How  many  yards  in  865  inches? 

17.  How  many  rods  in  1000  feet? 

18.  How  many  miles  in  2560  rods  ? 

19.  How  many  miles  in  3261  yards? 

20.  How  many  leagues  in  2365  rods  ? 

EXAMPLES    IN    REDUCTION    ASCENDING    AND    DESCENDING. 

1 29.  In  solving  the  following  examples,  the  pupil 
must  first  consider  whether  the  question  requires  hie/her 
denominations  to  be  reduced  to  lower,  or  lower  denomina- 
tions to  higher.  Having  settled  this  point,  he  will  find  no 
difficulty  in  applying  the  proper  rule. 

FEDERAL   MONEY.     (ART.  88.) 

1.  In  3  dollars  and  16  cents,. how  many  cents  ? 

2.  In  81  cents  and  2  mills,  how  many  mills? 

3.  In  245  cents,  how  many  dollars  ? 

4.  In  321  mills,  how  many  dimes? 

5.  In  95  eagles,  how  many  cents  ? 

6.  In  160  dollars,  how  many  cents  ? 

7.  In  317  dollars,  ho\v  many  dimes? 

8.  In  4561  mills,  how  many  dollars? 

9.  In  8250  cents,  how  many  eagles? 

10.  In  61  dolls.,  12  cts.,  and  3  mills,  how  many  mills? 


16  REDUCTION.  [SECT.   IX. 

STERLING  MONEY.     (ART.  90.) 

11.  Reduce  £17,  16s.  to  shillings. 

12.  Reduce  19s.  6d.  2  far.  to  farthings. 

13.  Reduce  1200  pence  to  pounds. 

14.  Reduce  3626  farthings  to  shillings. 

15.  Reduce  £19  to  farthings. 

16  Reduce  2880  farthings  to  shillings. 

17  Reduce  £21,  3s.  6d.  to  pence. 

18  Reduce  3721  farthings  to  pounds. 

TROY    WEIGHT.     (ART.  91.) 

19.  In  7  Ibs.,  how  many  ounces? 

20.  In  9  Ibs.  2  oz.,  how  many  pennyweights  ? 

21.  In  165  oz.,  how  many  pounds? 

22.  In  840  grains,  how  many  ounces? 

23.  In  3  Ibs.  5  oz.  2  pwts.  7  grs.,  how  many  grains  ? 

24.  In  6860  grains,  how  many  pounds? 

'     AVOIRDUPOIS   WEIGHT.     (ART.  92.) 

25.  In  200  oz.,  how  many  pounds  ? 

26.  In  261  Ibs.,  how  many  ounces?    ' 

27.  In  3  tons,  2  cwt.,  how  many  pounds? 

28.  In  1  cwt.  2  qrs.,  how  many  ounces? 

29.  In  1000  oz.,  how  many  pounds? 

30.  In  4256  Ibs.,  how  many  tons  ? 

APOTHECARIES'   WEIGHT.     (ART,  93.) 

31.  Reduce  45  pounds  to  ounces. 

32.  Reduce  71  oz.  to  scruples. 

33.  Reduce  93  Ibs.  2  oz.  to  grains. 

34.  Reduce  165  oz.  to  pounds. 

35.  Reduce  962  drams  to  pounds. 

LONG  MEASURE.     (ART.  94.) 
30.  In  636  inches,  how  many  yards  ? 
37.  In  763  feet,  how  many  rods? 


ART.  129.]  REDUCTION.  117 

38.  In  4  miles,  how  many  feet? 

39.  In  18  rods  2  feet,  how  many  inches? 

40.  In  1760  yards,  how  many  miles? 

41.  In  3  leagues,  2  miles,  how  many  inches? 

CLOTH   MEASURE.     (ART.  95.) 

42.  How  many  yards  in  19  quarters? 

43.  How  many  quarters  in  21  yards  and  3  quarters  ? 

44.  How  many  nails  in  35  yards  and  2  quarters? 

45.  How  many  Flemish  ells  in  50  yards  ? 

46.  How  many  English  ells  in  50  yards? 

47.  How  many  French  ells  in  50  yards  ? 

SQUARE  MEASURE.  (ART.  96.) 

48.  In  65  sq.  yards  and  7  feet,  how  many  feet  ? 

49.  In  39  sq.  rods  and  15  yds.,  how  many  yards  ? 

50.  In  27  acres,  how  many  square  feet? 

51.  In  345  sq.  rods,  how  many  acres? 

52.  In  461  square  yards,  how  many  rods? 

53.  In  876  sq.  inches,  how  many  sq.  feet? 

CUBIC  MEASURE.     (ART.  97.) 

54.  In  48  cubic  yards,  how  many  feet  ? 

55.  In  54  cubic  feet,  how  many  inches  ? 

56.  In  26  cords,  how  many  cubic  feet  ? 

57.  In  4230  cubic  inches,  how  many  feet? 

58.  In  3264  cubic  feet,  how  many  cords  ? 

WINE   MEASURE.    (ART.  98.) 

59.  Reduce  94  gallons  2  qts.  to  pints. 

60.  Reduce  68  gallons  3  qts.  to  gills. 

61.  Reduce  10  hhds.  15  gallons  to  quarts. 

62.  Reduce  764  gills  to  gallons. 

63.  Reduce  948  quarts  to  hogsheads. 

64.  Reduce  896  gills  to  gallons. 


118  REDUCTION.  [Se^r.  IX. 

BEER  MEASURE.     (ART.  99.) 

65.  How  many  quarts  in  11  hogsheads  of  beer? 

66.  How  many  pints  in  110  gallons  2  qts.  of  beer  ? 

67.  How  many  hogsheads  in  256  gallons  of  beer? 

68.  How  many  barrels  in  320  pints  of  beer? 

69.  How  many  pints  in  46  hlids.  10  gallons  ? 

70.  How  many  hhds.  in  2592  quarts  ? 

DRY  MEASURE.     (ART.  100.) 

71.  In  156  pecks,  how  many  bushels  ? 

72.  In  238  quarts,  how  many  bushels  ? 

73.  In  360  pints,  how  many  pecks  ? 

74.  In  58  bushels,  3  pecks,  how  many  pecks  ? 

75.  In  95  pecks,  2  quarts,  how  many  quarts  ? 

76.  In^YS  quarts,  how  many  bushels? 

77.  In  100  bushels,  2  pecks,  how  many  pints? 

TIME.     (ART.  101.) 

78.  How  many  minutes  in  16  hours  ? 

79.  How  many  seconds  in  1  day? 

80.  How  many  minutes  in  365  days  ? 

81.  How  many  days  in  96  hours  ? 

82.  How  many  days  in  3656  minutes  ? 

83.  How  many  seconds  in  1  week  ?     • 

84.  How  many  years  in  460  weeks? 

CIRCULAR  MEASURE.     (ART.  105.) 

85.  Reduce  23  degrees,  30  minutes  to  minutes. 

86.  Reduce  41  degrees  to  seconds. 

87.  Reduce  840  minutes  to  degrees. 

88.  Reduce  964  minutes  to  signs. 

o 

89.  Reduce  2  signs  to  seconds. 

90.  Reduce  5  signs,  2  degrees  to  minutes. 

91.  Reduce  960  seconds  to  degrees. 

92.  Reduce  1800  minutes  to  signs. 


ART.  117.]  REDUCTION.  119 

93.  In  45  guineas,  how  many  farthings  ? 

94.  In  60  guineas,  how  many  pounds  ? 

95.  In  62564  pence,  how  many  guineas  ? 

96.  In  £84,  how  many  guineas  ? 

97.  How  many  grains  Troy,  in  46  Ibs.  7  oz.  5  pwts.  ? 

98.  How  many  pounds  Troy,  in  825630  grains  ? 

99.  Reduce  62  Ibs.  10  pwts.  to  grains. 

100.  In  16  tons,  11  cwt.  9  Ibs.,  avoir.,  how  many  pounds  ? 

101.  Reduce  782568  ounces  to  tons. 

102.  In  18  rods,  2  yds.  3  ft.  10  in.,  how  many  inches  ? 

103.  How  many  feet  in  3  leagues,  2  miles,  12  rods  ? 

104.  In  2738  inches,  how  many  rods  ? 

105.  In  2  tons,  3  cwt.  2  qrs.  15  Ibs.,  how  many  ounces  ? 

106.  Reduce  53  Ibs.  11  pwts.  10  grs.  Troy,  to  grains. 

107.  How  many  English  ells  in  45  yards  ? 

108.  How  many  yards  in  45  English  ells  ? 

109.  How  many  Flemish  ells  in  54  yards  ? 
*     110.  How  many  French  ells  in  60  yards  ? 

111.  In  13  m.  2  fur.  6  ft..  7  in.,  how  many  inches  ? 

112.  In  84256  feet,  how  many  leagues  ? 

113.  In  135  bu.  3  pks.  2  qts.  1  pt.  how  many  pints  1 

114.  In  84650  pints,  how  many  quarters  ? 

115.  How  many  gills  in  48  hhds.  18  gal.  wine  measure  ? 

116.  How  many  pipes  in  98200  quarts? 

117.  How  many  seconds  in  15  solar  years  ? 

118.  How  many  weeks  in  8029200  seconds? 

119.  How  many  square  feet  in  82  acres,  36  rods,  8  yds.  ? 

120.  How  many  cords  of  wood  in  68600  cubic  inches  ? 

121.  How  many  inches  in  10  cords  and  6  cubic  feet  ? 

122.  In  246  tons  of  round  timber,  how  many  inches  ? 

123.  In  65200  square  yards,  how  many  acres? 

124.  In  8  signs,  43  deg.  18  sec.,  how  many  seconds  ? 

125.  In  75260  minutes,  how  many  signs  ? 


120  COMPOUND    ADDITION.  [SECT.  VII. 

COMPOUND  ADDITION. 

ART.   129*    Compound  Addition   is  the   process  of 
uniting  two  or  more  compound  numbers  in  one  sum. 

Ex.  1.  What  is  the  sum  of  £2,  3s.  4d.  1  far.;  £1,  6s. 
9d.  3  far.  ;  £7,  9s.  Yd.  2  far. 

Suggestion.  —  First    write    the  Operation. 

numbers  under  each  other,  pounds  £      St     rft   yart 

under  pounds,  shillings  under  shil-  2  "  4  "  4  "  1 

lings,  <fec.     Then,  beginning  with  1  "  6  '   9      3 

the  lowest  denomination,  we  find  '       ^       * 


the  sum  is  6  farthings,  which  is   -ns.  11  "  0  "  9  "  2 
t^ua]  to  1  penny  and  2  far.  over.     Write  the  2  far.  under 
the  column  of  farthings,  and  carry  the  Id.  to  the  column  of 
pence.     The  sum  of  the  pence  is  21,  which  is  equal  to  Is. 
and  9d.     Place  the  9d.  unda    the  column  of  pence,  and 
carry  the  Is.  to  the  column  of  shillings.     The  sum  of  the 
shillings  is  20,  which   is  equal  to   £l   and   nothing  oveiv, 
Write  a  cipher  under  the  column  of  shillings,  and  carry  the 
£1  to  the  column  of  pounds.     The  sum  of  the  pounds  is 
11,  which  is  set  down  in  full. 

13O*  Hence,  we  derive  the  following  general 

RULE   FOR   COMPOUND   ADDITION. 

I.  Write  the  numbers  so  that  the  same  denominations 
shall  stand  under  each  other. 

II.  Beginning  at  the  right  hand,  add  each  column  sepa- 
rately, and  divide  its  sum  by  the  number  required  to  make 
ONE  of  the  next  higher  denomination.     Setting  the  remain- 
der under  the  column  added  ,  carry  the  quotient  to  the  next 
column,  and  thus  proceed  as  in  Simple  Addition.    (Art.  23.) 

PROOF.  —  The  proof  is  the  same  as  in  Simple  Addition. 

QUEST.—  129.  What  is  Compound  Addition?  130.  How  do  you  write  com- 
pound numbers  for  addition  ?  Where  do  you  begin  to  add,  and  how  pro- 
ceed ?  How  is  Compound  Addition  proved  ? 


ARTS.  129,130.]    COMPOUND  ADDITION.  121 


(2.) 
£  8.  d.  far. 

1362 
3  08  3 
9  18  9  1 

(3.) 
Ib.  oz.  pwt.  gr. 
2574 
2  0  5  19 
6803 

(*•) 
m.  r.  ft.  in. 

7  15  20  8 
6487 
9644 

U     3     0     2  Arts.  11      i   13     2Ans.  22  27     0     T  Ans 

(6.)  (6.)  (7.) 

£     5.    d.  /ar.  /&.  oz.  pwt.  gr.  r.  yc/.  ft.   in. 

10  17     0     1  17  10  13     5  4426 

19  652     8928  6602 

7820  10  4  11  3  6814 

3263  21  11  16  6  2325 

(8.)  (9.)           (10.) 

cwt.  qr.  Ib.  oz.  wk.  d.  hr.  min.  yd.  qr.  na.  in. 

5345  13  4  19  30  6312 

6298  15  13  16  3231 

8172  73   5  10  7024 

6096  12  0  14  25  5112 

11.  Add  4  tons,  5  cwt.   3   qrs.  2  Ibs.  10  oz.  4  drs. ;  6 
tons,  4  cwt.  17  Ibs.  15  oz.  9  drs. ;  3  tons,  2  cwt.  1  qr.  15  Ibs. 

12.  Add  4  hhds.  10  gals.  3  qts.  1  pt. ;  15  hhds.  19  gals. 

2  qts. ;   8  hhds.  7  gals.  2  qts.  1  pt.  wine  measure. 

13.  Add  1  pipe,  1  hhd.  8  gals.  2  qts.  1  pt.  2  gills  ;    1 
pipe,  6  gals.  1  qt. ;  3  pipes,  1  hhd.  3  gals.  3  qts.  1  pint.   . 

14.  A  man  sold  the  following  quantities  of  wheat :  5  bu. 

3  pks.  2  qts. ;   10  bu.  1  pk.  4  qts. ;  21  bu.  2  pks.  5  qts. : 
how  much  did  he  sell  in  all  ? 

15.  A  merchant  bought  3  pieces  of  silk,  one  of  which 
contained  21  yds.  2  qrs.  3  nails  ;  another  19  yds.  3  qrs.  1 
nail ;  and  the  other  26  yds.  1  qr.  and  2  nails :  how  many 
yards  did  they  all  contain  ? 

5 


122  COM  I -oi  :  ACTION.  [SECT.  VIIL 

COMPOUND  SUBTRACTION. 

ART.  131*   Compound  Subtraction  is  the  process  of 
finding  lh<-  difference  between  two  compound  n umbel's. 
i:\.  1.  Fn.in  £11,  8s.  5d.  3  far.,  subtract  £5,  10s.  2d.  1 

farthing. 

» 

Suggestion. — Write  the  less  number         Operation. 
under  the  greater,  pounds  under  pounds,      £      8.    d.far. 
shillings  uml.T  shillings,  <fec.     Then,  be-     11      8     5     3 

ginning  with  tin-   lowest  denomination,        5   10     2 1^ 

proceed  thus  :  1  far.  from  3  far.  leaves  2  5  18  3  2 
t'.ir.  Set  the  reiiiaind-r  2  under  the  farthings.  Next,  2d. 
frmnod.  leave  3d.  Write  the  3  under  the  pence.  Since 
10  shillings  cannot  be  taken  from  8  shillings  ;  we  b< 
as  many  shillings  as  it  takes  to  make  one  of  the  next 
highrr  denomination,  which  is  pounds;  and  £1,  or  20s., 
add. -d  to  the  8s.  make  28  shillings.  Now  10s.  from  28s. 
lea^e  18s.,  which  we  write  under  the  shillings.  Finallv, 
carrying  1  to  the  next  number  in  the  lower  line,  we  have 
£0  ;  and  £6  from  £11  leave  £5,  which  we  write  under 
the  pounds.  The  answer  is  £5,  18s.  3d.  2  far. 

132*   Hence,  we  derive  the  following  general 

KULK   FOR  COMPOUND  SUBTRACTION. 

I.  Write  the  less  number  under  the  greater,  so  that  the 

same  denominations  may  stand  under  each  other. 

II.  Beginning  at  the  right  hand,  subtract  each  lower 
number  from  the  number  above  it,  and  set  the  remainder 
under  tie  number  subtracted. 

III.  When  a  number  in   the  lower  line  is  larger  than 
that  above  it,  add  as  many  units  to  the  upper  number  an  it 

QUEST.— 131.  What  is  Compound  Subtraction  1     132.   How  do  you   write 

compound  numbers  for  subtraction?  Where  begin  to  subtract,  and  how 
proo'ed  ?  When  a  number  io  the  lower  line  is  lanriT  tlmn  that  above  it. 
whnt  is  to  be  done  ? 


ARTS.  181,  132.]  COMPOUND  SUBTRACTION.  123 

takes  to  make  ONE  of  the  next  higher  denomination  ;  then 
subtract  as  before,  and  adding  1  to  the  next  number  in  the 
lower  line,  proceed  as  in  Simple  Subtraction. 

PROOF. — The  proof  is  the  same  as  in  Sim.  Subtraction. 

(2.)  (3.) 

From  £13,  7s.  8d.  3  far.  19  Ibs.  3  oz.  7  pwts.  12  grs. 
Take  £  6,  5s.  lid.  1  far.  15  Ibs.  8  oz.  3  pwts.  4  grs. 

(4.)  (5.) 

From  12  T.  7  cwt.  1  qr.  3  Ibs.  15  m.  3  fur.  10  r.  8  ft.  4  in. 
Take  7  T.  9  cwt.  3  qrs.  4  Ibs.  9  m.  6  fur.  3  r.  4  ft.  7  in. 

6.  From  24  yds.  2  qrs.  3  nails,  take  16  yds.  3  qrs.   2 
nails. 

7.  A  lady  having  £18,  4s.  7d.  in  her  purse,  paid  £8,  7s. 
3d.  for  a  dress  :  how  much  had  she  left  ? 

8.  If  from  a  hogshead  of  in  u  draw  out  19  gals 

3  qts.  1  pt,  how  much  will  there  be  left  in  the  hogshead 

9.  A  person  bought  8  tons,  3  cwt.  19   Ibs.  of  coal,  and 
having  burned  3  tons,  6  cwt.  45  Ibs.  sold  the  rest:   how 
much  did  he  sell  ? 

10.  From  17  years,  7  mos.  16  days,  take  15  years,  and 

4  months. 

11.  From  39  yrs.  3  mos.  7  days,  4  min.,  take  23  yrs.  5 
mos.  3  days,  16  bra. 

12.  From  43  A.  2  roods,  15  rods,  take  39  acres  and  11 
rods. 

13.  From  38  leagues,  2  miles,  5  fur.  17  rods,  take  29 
leagues,  2  miles,  7  fur.  13  rods. 

14.  From  125  bushels,  3  pecks,  4  quarts,  2  pints,  take 
108  bushels,  2  pecks,  7  quarts. 

15.  From  85  guineas,  13  shillings,  4  pence,  2  far.  take 
39  guineas,  15  shillings,  8  pence. 

QUBST.— How  is  Compound  Subtraction  orovedl 


124  COMPOUND    MUI/H1>UCATION.        [SECT.   VLII. 

COMPOUND  MULTIPLICATION. 

ART.  133*  Compound  Multiplication  is  the  process 
of  finding  the  amount  of  a  compound  number  repeated  or 
added  to  itself,  a  given  number  of  times. 

Ex.  1.  What  will  3  barrels  of  flour  cost,  at  £l,  7s.  5d.  2 
far.  per  barrel  ? 

Suggestion. — Write  the  multiplier  un-        - 

der  the  lowest  denomination  of  the  multi- 

...  Jo     ft.    a.  jar. 

pncand,  and  proceed  thna  :  3  times  2  far.     i  "  7  "  .-, 

are  6   far.  which   are  equal  to   Id.  and  2  3 

far.    o\vr.       Write  the  remainder    2    fur.     4 £     J     o 
under  the  denomination   multiplied,  and 
carry  the  Id.  to  the  next  product     3  times  5d.  are  ir>d., 
and    1    to  carry  makes    10d.,  equal   to   Is.  and   4d.  over. 
Write  the  4d.  under  the  pence,  and  carry  the  Is.  to  the 
next  product.     r.i  times  7s.  are  21s.  and  1  to  carry  n. 
22s.,  equal  to  £l,  and  2s.     Write  the  2  under  the  shillings 
ana  carry  the  £1   to  the  next  product.     Finally,  3  times 
£1  are  £3,  and  1  to  carry  makes  £4.     Write  the  £4  under 
the  pounds.     The  answer  is  £4,  2s.  4d.  2  far. 

134*  Hence,  we  derive  the  following  general 

RULE   FOR   COMPOUND    MULTIPLICATION". 
Beginning  at  the  right  hand,  multiply  each  denomina 
t ion  of  the  multiplicand  by  the  multiplier  separately,  and 
divide  its  product  by  the  number  required  to  make  ON 
the  next  higher  denomination,  setting  down  the  remainder 
and  carrying   the   quotient    as   in    Compound  Addition. 

2.  Multiply  £4,  6s.  2d.  3  far.  by  15. 

3.  Multiply  19  Ibs.  8  oz.  9  pwts.  7  grs.  by  12. 

4.  If  a  man  walks  3  mile*,  3  fur.  18  rods  in  1  hour,  h.»\v 
far  will  he  walk  in  1 0  hours  ? 

QUEST.— 133.  Wh:it  Is  Compound  Multiplication  1     (34.  What  is  the  rule 
fi<r  <'imifw»MMf1  Multiplicutiou  ? 


ARTS.  133,  134.]       COMPOUND  DIVISION.  125 

5.  Multiply  7  leagues,  1  m.  31  rods,  12  ft.  3  in.  by  9. 

6.  Multiply  18  tons,  3  cwt.  10  Ibs.  7  oz.  3  drs.  by  11. 

7.  A  man  has  7  pastures,  each   containing   6  acres,  25 
rods,  51  square  feet:  how  much  do  they  all  contain? 

8.  A  man    bought  9  loads  of  wood,  each    containing    L 
cord  and  21  cu.  ft.  :  how  much  did  they  all  contain  \ 

9.  Multiply  17  yds.  3  qrs.  2  nails  by  35. 

10.  Multiply  53  days,  19  min.  7  sec.  by  41. 

11.  Multiply  30  years,  3  weeks,  5  days,  12  hours,  by  03. 

12.  Multiply  G5  hhd.s.  23  gals.  3  qts.  1  pt.  by  72. 

COMPOUND  DIVISION. 


135.    Compound  Diris'nn    i>   the  process  of  dividing 

nuinhcis. 


Ex.  1.  A   father  divided   £10,   5s.   8d.   2   far.  equally 
among  his  3  sons  :  how  much  did  each  receive  ? 

Suggestion. — Write  the  divi->r 

i      ,  ,.      «    ,       ,.  . ,  Operation. 

on   the   left  of  the  dividend,  and  „  . 

proceed  as  in  Short  Division.  Thus,         $\\Q  ,/  £  ,/  g",/ ^  " 

3  is  contained  in  £10,  3  times  and  o  //  o  ///»//» 

lt.  -_T         .  -4W5.  o       o       b       o* 

JL1  over.     We  write  the  3  under 

the  pounds,  because  it  denotes  pounds  ;  then  reducing  the 
remainder  £l  to  shillings  and  adding  the  given  shillings  5, 
we  have  25s.  Again,  3  is  in  25s.  8  times  and  Is.  over.  We 
set  the  8  under  the  shillings,  because  it  denotes  shillings  ; 
then  reducing  the  remainder  Is.  to  pence  and  adding  the 
given  pence  8,  we  have  20d.  Now  3  is  in  20d.  0  tim-.-s 
and  2d.  over.  We  set  the  6d.  under  the  pence,  because  it 
denotes  pence.  Finally,  reducing  the  rem.  2d.  to  farthings 
and  adding  the  given  far.  2,  we  have  10  far. ;  and  3  is  in 
10,  3  times  and  1  far.  over.  Write  the  3  under  the  far 

• >*OT    T 

QUKBT.— 135.  What  is  Compound  Division  ? 


126  COMPOUND  DIVISION.  [SECT.  VIII. 

1 36.  Hence,  we  derive  the  following  general 

RULE   FOR   COMPOUND   DIVISION. 

1.  JJr(/ii'tnhtfj  at  the  left  hand,  divide  each  denomination 
of  the  dividend  by  the  dirixor,  and  write  the  quotient  fig- 
ures under  the  figures  divided. 

II.  If  there  is  a  remainder,  reduce  it  to  the  next  lower 
jLcnomination,  and  adding  it  to  the  figures  of  the  corn  .^pond- 
ing denomination  of  the  divide  ml,  i/iri'/c  this  n  timber  as 
before.  Thus  proceed  through  all  the  denomiMtiOM)  ami 
the  several  quotients  will  be  the  answer  required. 

OBS.  1.  Each  quotient  figure  is  of  the  same  denomination  as  that 
part  of  the  dividend  from  which  it  arose. 

2.  When  the  divisor  exceeds  12,  and  is  a  composite  number,  we 
may  divide  first  by  one  factor  and  that  quotient  by  the  other. 

2.  Divide  14  Ibs.  5  oz.  G  pwts.  9  grs.  by  3. 

3.  Divide  £5,  17s.  8.1.  1  far.  by  4. 

4.  Divide  25  Ibs.  3  ounces,  8  pwts.  7  grs.  by  5. 

5.  Divide  15  T.  15  cwt.  3  qrs.  10  Ibs.  by  6. 

6.  Divide  23  yards,  2  qrs.  1  nail,  by  7. 

7.  Divide  35  leagues,  1  in.  3  fur.  17  rods  by  8. 

8  Divide  45  hhds.  18  gals.  39  qts.  1  pint  by  9. 

9  A  fanner  had  34  bu.  3  pks.  1  qt.  of  wheat  in  9  bags  : 
ow  much  was  in  each  bag? 

10.  If  you  pay  £25,  17s.  8-j-d.  for  5   cows,  how  much 
will  that  be  apiece  ? 

1 1.  Divide  38  tons,  5  cwt.  2  qrs.  15  Ibs.  by  17. 

12.  Divide  41  hhds.  13  gals.  2  qt.  wine  measure  by  23. 

13.  Divide  54  acres,  2  roods,  25  rods,  by  34. 

14.  Divide  29  cords,  19  cu.feet,  18  cu.  inches  by  41. 

15.  Divide  78  years,  17  weeks,  24  days,  by  63. 


QITKKT. — 136.  What  is  the  rule  for  Compound  Division  ?     Obs.  Of  what  do- 
uoumuiioii  is  each  quota-lit  llgure  7 


MISCELLANEOUS    EXERCISES.  127 

MISCELLANEOUS  EXERCISES. 

1.  From  the  sum  of  463279  +  734658,  take  926380. 

2.  To  the  difference  of  8.56273  and  46719,  add  4203,76. 

3.  To  476208  add  5207568  —  4808345. 

4.  Multiply  the  sum  of  863576  +  435076  by  287. 

5.  Multiply  the  difference  of  870358  —  640879  by  365. 

6.  Divide  the  sum  of  439409  +  87646  by  219. 

7.  Divide  the  difference  of  607840  — 23084  by  367. 

8.  Divide  the  product  of  865060X406  by  1428. 

9.  Divide  the  quotient  of  55296+144  by  89. 

10.  What  is  thesum  of  4845  +  76  +  1009  +  463+407  ? 

1 1.  What  is  the  sum  of  836  X  46,  and  784  X  76  ? 

12.  What  is  the  sum  of  1728+72,  and  2828-7-96? 

13.  What  is  the   sum  of  85263  — 45017,  and  68086? 

14.  What  is  the  difference  between  38076  +  16325,  and 
20268  +  45675  ? 

15.  W7hat  is  the  difference  between  40719  +  6289,  and 
31670  —  18273. 

16.  What  is  the  difference  between  378X  96,  and  9419  > 

17.  What  is   the  difference  between  7560-7-504,  and 
7560X504? 

18.  Froml45X87,  take  12702+87. 

19.  Multiply  83X19  by  75X23. 

20.  How  many  times  can  34    be  subtracted  from  578? 

21.  How  many  times  can  1512  be  taken  from  7569  ? 

22.  How  many  times  can  63  X  24  be  taken  from  27640  \ 

23.  How  many  times  is  68  +  31  contained  in  45600? 

24.  Divide  832  +  1429  by  45  +  84. 

25.  Divide  467+2480  by  346  —  187. 

26.  Divide  68240  —  16226  by  10405  —  6200. 

27.  Divide  320X160  by  2125  —  960. 

28.  Divide  826340  —  36585  by  126X84. 

29.  From  62345  +  19008,  take  2134X38. 

30.  From  2631  X216,  take  576+36. 


123  MISCELLANEOUS    EXERCISES. 

33.  A  young  man  having  50  dollars,  bought  a  coat  for 
15  dollars,  a  pair  of  pants  for  8  dollars,  a  vest  for  5  dol- 
lars, and  a  hat  for  3  dollars :  how  much  money  did  he 
have  left  ? 

34.  A  farmer  sold  a  cow  for  18  dollars,  a  calf  for  4 
dollars,  and  a  lot  of  sheep   for  35  dollars :  how  much 
more  did  he  receive  for  his  sheep  than  for  his  cow  and 
calf? 

35.  A  man  having  90  dollars  in  his  pocket,  paid  27 
dollar  for  9  cords  of  wood,  35  dollars  for  7  tons  of  coal, 
and  11  dollars  for  carting  both  home:  how  much  money 
had  he  left  ? 

36.  A  young  lady  having  received  a  birthday  present 
of  100  dollars,  spent  17  dollars  for  a  silk  dress,  26  dol- 
lars for  a  crape  shawl,  and  8  dollars  for  a  bonnet :  how 
many  dollars  did  she  have  left  ? 

37.  A  dairy- woman  sold   23  pounds  of  butter  to  one 
customer,  34  pounds  to  another,  and  had  29  pounds  left: 
how  many  pounds  had  she  in  all  ? 

38.  A  lad  bought  a  pair  of  boots  for  16  shillings,  a 
pair  of  skates  for  10  shillings,  a  cap  for  17  shillings,  and 
had  20  shillings  left:  how  many  shillings  had  he  at  first? 

39.  A  grocer  having  500  pounds  of  lard,  sold  3  kegs 
of  it ;  the  first  keg  contained  43  pounds,  the  second  45 
pounds,  and  the  third  56  pounds  :  how  many  pounds  did 
he  have  left? 

40.  A  man  bought  a  horse  for  95  dollars,  a  harness 

o 

for  34  dollars,  and  a  wagon  for  68  dollars,  and  sold  them 
all  for  225  dollars  :  how  much  did  he  make  by  his  bar- 
gain ? 

41.  A  person  being  1000  miles  from  home,  on  his  re- 
turn, traveled  150  miles  the  first  day,  240  miles  the  sec- 
ond day,  and  310  miles  the  third  day:  how  far  from 
home  was  he  then  ? 


MISCELLANEOUS    EXERCISES.  129 

42.  George  bought  a  pony  for  78  dollars  and  paid  3 
dollars  for  shoeing  him;  he  then  sold  him  for  100  dol- 
lars :  how  much  did  he  make  by  his  bargain  ? 

43.  A  man  bought  a  carriage  for  273  dollars,  and  paid 
27  dollars  for  repairing  it ;  he  then  sold  it  for  318  dol- 
lars: how  much'did  he  make  by  his  bargain? 

44.  A  man  bought  a  lot  for  275  dollars,  and  paid  a 
carpenter  850  dollars  for  building  a  house  upon  it :  he 
then  sold  the  house  and  lot  for  1200  dollars:  how  much 
did  he  make  by  the  operation  ? 

45.  A  farmer  having  150  sheep,  lost  17  and  sold  65 ; 
he  afterwards  bought  38  :  how  many  sheep  had  he  then  ? 

46.  A  man  bought  27  cows,  at  31  dollars  per  head: 
how  many  dollars  did  they  all  cost  him  ? 

47.  A  miller  sold  251  barrels  of  flour,  at  8  dollars  a 
barrel :  how  much  did  it  come  to  ? 

48.  A  merchant  sold  218  yards  of  cloth,  at  8  dollars 
per  yard :  how  much  did  it  come  to  ? 

49.  A  merchant  sold  18  yards  of  broadcloth,  at  4  dol- 
lars a  yard,  and  21  yards  of  cassimere,  at  2  dollars  a  yard : 
how  much  did  he  receive  for  both  ? 

50.  A  farmer  sold  12  calves,  at  5  dollars  apiece,  and 
35  sheep,  at  3  dollars  apiece :  how  much  did  he  receive 
for  both  ? 

51.  A  grocer  sold  to  one  person  25  firkins  of  butter, 
at  7  dollars  a  firkin,  and  13  to  another,  at  8  dollars  a  fir- 
kin :  how  much  did  both  lots  of  butter  come  to  ? 

52.  A  shoe  dealer  sold  100  pair  of  coarse  boots  to  one 
customer,  at  4  dollars  a  pair,  and  156  pair  of  fine  boots 
to  another,  at  5  dollars  a  pair:  what  did  both  lots  of 
boots  come  to  ? 

53.  A  miller  bought  165  bushels  of  corn,  at  5  shillings 
a  bushel,  and  286   bushels   of  wheat,  at   9  shillings  a 
bushel :  how  much  did  he  pay  for  both  ? 


130  MISCELLANEOUS    EXERCISES. 

54.  A  man  bought  45  clocks,  at  3  dollars  apiece,  and 
sold  them,  at  5  dollars  apiece :  how  much  did  he  make  by 
his  bargain  ? 

55.  A  bookseller  bought  87  books,  at  7  shillings  apiece, 
and  afterwards  sold  them,  at  6  shillings  apiece :  how  much 
did  he  lose  by  the  operation  ? 

56.  How  many  yards  of  calico,  at  18  cents  a  yard,  can 
»e  bought  for  240  cents  ? 

57.  A  little  girl  having  326  cents,  laid  it  out  in  ribbon, 
at  25  cents  a  yard :  how  many  yards  did  she  buy  ? 

58.  If  a  man  has  500  dollars,  how  many  acres  of  land 
can  he  buy,  at  15  dollars  per  acre? 

59.  How  many  cows,  at  27  dollars  apiece,  can  be  bought 
for  540  dollars  ? 

60.  How  many  barrels  of  sugar,  at  23  dollars  per  bar- 
rel, can  a  grocer  buy  for  575  dollars? 

61.  Henry  sold  his  skates  for  87  cents,  and  agreed  to 
take  his  pay  in  oranges,  at  3  cents  apiece :    how  many 
oranges  did  he  receive  ? 

62.  William  sold  80  lemons,  at  4  cents  apiece,  and  tool 
his  pay  in  chestnuts,  at  5  cents  a  quart :  how  many  chest- 
nuts did  he  get  for  his  lemons  ? 

63.  A  milkman  sold  110  quarts  of  milk,  at  6  cents  a 
quart,  and  agreed  to  take  his  pay  in  maple  sugar,  at  11 
cents  a  pound  :  how  many  pounds  did  he  receive  ? 

64.  A  farmer  bought  25  yards  of  cloth,  which  was 
worth  6  dollars  per  yard,  and  paid  for  it  in  wood,  at  2 
dollars  per  cord :  how  many  cords  did  it  take  ? 

65.  A  pedlar  bought  4£  pieces  of  silk,  at  24  dollars 
apiece :  how  much  did  he  pay  for  the  whole  ? 

66.  A  farmer  sold  8-}-  bushels  of  wheat,  at  96  cents 
per  bushel :  how  much  did  he  receive  for  his  wheat  ? 

67.  A  man  sold  a  lot  of  land  containing  15^  acres,  at 
i  f\  dollars  per  acre :  how  much  did  he  receive  for  it  ? 


MISCELLANEOUS    EXERCISES.  131 

68.  If  a  man  can  walk  45  miles  in  a  day,  how  far  can 
he  walk  in  20f  days  ? 

69.  What  cost  75  yds.  of  tape,  at  •£  of  a  cent  per  yd.  ? 

70.  What  will  100  pair  of  childrens*  gloves  come  to, 
at  -ft  of  a  dollar  a  pair  ? 

71.  What  will  160  boys'  caps  cost,  at  -f  of  a  dollar 
apiece  ? 

72.  What  will  210  pair  of  shoes  cost,  at  f  of  a  dollar 
a  pair  ? 

73.  How  many  childrens'  dresses  can  be  made  from  a 
piece  of  lawn  which  contains  54  yards,  if  it  takes  4£  yards 
for  a  dress  ? 

74.  A  farmer  wishes  to  pack    100  dozen  of  eggs  in 

f  OO 

boxes,  and  to  have  each  box  contain  6-}-  dozen :  how  many 
boxes  will  he  need  ? 

75.  A  lad  having.  275  cents,  wishes  to  know  how  many 
miles  he  can  ride  in  the  Railroad  cars,  at  2jr  cents  per  mile : 
how  many  miles  can  he  ride  ? 

76.  How  many  apples,  at  £  a  cent  apiece,  can  Horatio 
buy  for  75  cents? 

77.  If  Joseph  has  to  pay  f  of  a  cent  apiece  for  marbles, 
how  many  can  he  buy  for  84  cents  ? 

78.  At  -f  of  a  dollar  apiece,  how  many  parasols  can  a 
shopkeeper  buy  for  168  dollars? 

79.  If  I  am  charged  -f-  of  a  dollar  apiece  for  fans,  how 
many  can  I  buy  for  265  dollars  ? 

80.  How  many  yards  of  silk,  which  is  worth  -£$  of  a 
dollar  a  yard,  can  I  buy  for  227  dollars  ? 

81.  How  many  pair  of  slippers,  at  -f  of  a  dollar  a  pair, 
can  be  bought  for  448  dollars  ? 

82.  In  £45,  13s.  6d.,  how  many  pence? 

83.  In  £63,  7s.  8d.  2  far.,  how  many  farthings? 

84.  How  many  yards  of  satin  can  I  buy  for  £75,  10s., 
if  I  have  to  pay  5  shillings  per  yard  ? 


132  MISCELLANEOUS    EXERCISES. 

85.  How  many  six-pences  are  there  in  £100  ? 

86.  A  grocer  sold  10  hogsheads  of  molasses,  at  3  shil- 
lings per  gallon :  how  many  shillings  did  it  come  to  ? 

87.  A  milkman  sold  125  gallons  of  milk,  at  4  cents  per 
quart :  how  much  did  he  receive  for  it  ? 

88.  A  man  made  30  barrels  of  cider  which  he  wished  to 
put  into  pint  bottles :  how  many  bottles  would  it  require  ? 

89.  How  much  would  85  bushels  of  apples  cost,  at  12 
cents  a  peck  ? 

90.  What  will  97  pounds  of  snuff  cost,  at  8  cents  per 
ounce  ? 

91.  What  will  5  tons  of  maple  sugar  come  to,  at  11 
cents  a  pound  ? 

92.  A  farmer  sold  34  tons  of  hay,  at  65  cents  per  hun- 
dred :  how  much  did  he  receive  for  it  ? 

93.  A  blacksmith  bought  53  tons  of  iron  for  3  'lollars 
per  hundred :  how  much  did  he  pay  for  it  ? 

94.  A  young  man  returned  from  California  with  50 
pounds  of  gold  dust,  which  he  sold  for  16  dollars  per 
ounce  Troy :  how  much  did  he  receive  for  it  ? 

95.  A  man  bought  36  acres  of  land  for  3  dollars  per 
square  rod :  how  much  did  his  land  cost  him  ? 

96.  John  Jacob  Astor  sold  five  building  lots  in  the  city 
of  New  York,  containing  560  square  rods,  for  13  dollars 
per  square  foot :  how  much  did  he  receive  for  them  ? 

97.  A  laboring  man  engaged  to  work  5  years  for  16 
dollars  per  month :  what  was  the  amount  of  his  wages  ? 

98.  What  will  17   cords  of  wood  cost,  at  6  cents  pei 
cubic  foot  ? 

99.  If  it  takes  35  men  18  months  to  build  a  fort,  how 
many  years  would  it  take  1  man  to  build  it  ? 

100.  If  it  takes  1  man  360  days  to  build  a  house,  how 
many  weeks  would  it  take  15  men  to  build  it,  allowing  6 
working  days  to  a  week? 


ANSWERS  TO  EXAMPLES. 


133 


ANSWERS  TO  EXAMPLES. 
ADDITION. 


Ex.              Ans. 

Ex.               Ans. 

Ex.               Ans. 

ART.  2O. 

4.  5286  yards. 

28.  171658. 

1.  Given. 

5.  2404. 

29.  57  dollars. 

2.  68. 

6.  2765. 

30.  58  dollars. 

3.  589. 

7.  10040. 

31.  120  dollars. 

4.  768. 

8.  8668. 

32.  565. 

5.  9987. 

9.  84  inches.  . 

33.  742. 

6.  878. 

10.  114  feet. 

34.   1530. 

V.  6767. 

11.  168  dollars. 

35.  1779. 

8.  8898. 

12.  192  rods. 

36.   1597. 

9.  8779. 

13.  782  pounds. 

37.   1757. 

10.  6796. 

14.   1380  yards. 

38.  2379. 

11.  88776. 

15.  576  miles. 

39.  2619. 

12.  986788. 

16.  836  sheep. 

40.  1020. 

17.  615  dollars. 

41.  1418. 

ART.  22. 

18.   181  dollars. 

42.  1191. 

13,  14.  Given. 

19.   1452. 

43.  150  bushels. 

15.   1454. 

20.   1255. 

44.  133  yards. 

16.  15300. 

21.  ,1881. 

45.  731  acres. 

37.  13285. 

22.  6693. 

46.  1197  cattle. 

23.  20485. 

47.   12554  dollars. 

ART.  24. 

24.  9726. 

48.   1282. 

1.  155  pounds.      25.  1769. 

49.  2528. 

2.  413  feet.            26.   1500. 

50.  365  days. 

3.  1960  dollars. 

27.  106284. 

ART.  2  4.  a.  . 

10.  65471. 

20.  551452. 

30.  279,075. 

1.  300. 

11.  327371. 

21.  46157. 

31.  295,306. 

2.  6000. 

12.  390497. 

22.  424634. 

32.  1,606,895. 

3.  9000. 

13.  37938. 

23.  430032. 

35.  6,140,704. 

4.  4861. 

14.  50342. 

24.  3458772. 

36.  7,569,904. 

5.  4871. 

15.  449458. 

25.  48350. 

37.  9,253,854. 

6.  47067. 

16.  466789. 

26.  514299. 

38.  9,247,176. 

7.  53340. 

17.  40290. 

27.  595522. 

39.  10,531,960. 

8.  59139. 

18.  50676. 

28.  5781566. 

40.  12,811,860 

9.  61304. 

19.  508302. 

29.  61993. 

134 


ANSWERS.  [PAGES  28 — 35. 


SUBTRACTION. 


Ex.              Ans. 

Ex.               Ans. 

Ex.               Ans. 

ART.  28. 

14.  275  pounds. 

48.  222  bushels. 

1.  Given. 

15.  613  yards. 

49.  195  dollars. 

2.  24. 

16.  310  rods. 

50.  1122  dollars. 

3.  12. 

17.  230  gallons. 

51.   1659  dollars. 

4.  153. 

18.  503  hhds. 

52.  3023  dollars. 

5.   245. 

19.  76  bushels. 

53.   1763  dollars. 

6.  31  dollars. 

20.   127  dollars. 

54.  3747  dollars. 

7.  12  pounds. 

21.  249  pounds. 

55.   16014  dollars. 

8.   115  yards. 

22.   1082  rods. 

56.   1315  dollars. 

9.  222  shillings. 

23.   13016. 

57.  5385  dollars. 

10.  222  marbles. 

24.  310768. 

58.   5735  dollars. 

25.  464374. 

59.   13944  soldiers 

ART.  3O. 

26.  5244038. 

60.  94760000  m. 

11,  12.  Given. 

27.  45. 

61.  17  oranges. 

13.  137. 

28.  308. 

62.  33  marbles. 

14.  2616. 

29.  240. 

63.  76  sheep. 

15.  3270. 

30.  58. 

64.  52  cents. 

16.  3203. 

31.  542. 

65.  43  yards. 

17.  5365667. 

32.  2021. 

66.  122  dollars 

33.   1825. 

67.  87  dollars. 

ART.  32. 

34.  2600. 

68.  66  pears. 

1.  217.* 

35.  3085. 

69.  59. 

2.  182. 

36.  1306. 

70.  164. 

3.  242. 

37.  4098. 

71.  149  pounds. 

4.  369. 

38.   1108. 

72.   164  bushels. 

5.  1029. 

39.  4531. 

73.  263  miles. 

6.  1008. 

40.  14520. 

74.  125  gallons. 

7.  3289. 

41.  24622. 

75.  179  pounds. 

8.  3434. 

42.   125028. 

76.   175  dollars. 

9.  35100. 

43.  64303. 

77.  339  pounds. 

10.  312657. 

44.  224066. 

78.   172  barrels. 

11.   1. 

45.   103875. 

79.  297  pages. 

12.  23  dollars. 

46.  420486. 

80.   110  dollars. 

13.  57  bushels. 

47.  72  sheep. 

81.  392  dollars. 

*  It  is  an  excellent  exercise  for  the  pupil  to  prove  all  the  examples.    This 
one  of  the  best  means  to  give  him  confidence  in  his  own  powers. 


I1  AGES  39 46.]  ANSWERS. 


135 


MULTIPLICATION. 


Ex.              Ans. 

Ex.              Ans. 

Ex.               Aus. 

ART.  39. 

ART.  41. 

33.  9100  weeks. 

1.  Given. 

34—37.  Given. 

34.  23760  min. 

2.  68: 
3.  936. 

ART.  43. 

35.  28350  gallons, 
36.  34675  dolls. 

4.   8084. 
5.   5550. 

1.   252. 
2.  390. 

37.  33840  sq.  in. 
38.  26070  miles. 

6.   12066. 

3.  567. 

7.  24408. 

4.  582. 

ART.  45. 

8.   35550. 

5.  840. 

40.  Given. 

9.  56707. 

6.   1155. 

41.  260. 

10.  Given. 

7.   3568. 

42.  3700. 

8.  2763. 

43.  51000. 

ART.  4O. 

9.  3920. 

44.  226000. 

11.   312. 

10.  460. 

45.  341000. 

12.  480. 

11.  572. 

46.  4690000. 

13.  249. 

12.  816. 

47.  52300000. 

14.  840. 

13.   1092. 

48.  681000000. 

15.   828. 

14.   1170. 

49.  856120000. 

16.  815. 

15.  2185. 

50.  96030500000. 

17.   2248. 

16.  4515. 

51.  Given. 

18.  3144. 

17.  12306. 

19.  2520. 

18.  25355. 

ART.  46. 

20.   1900. 

19.  342  dollars. 

52.   17000. 

21.  3960. 

20.  336  bushels. 

53.  291000. 

22.  6560. 

21.  336  inches. 

54.  4920000. 

23.  5628. 
24.  8712. 

22.  620  pounds. 
23.  391  dollars. 

55.   11700000. 
56.  33930. 

25.   1050  dollars. 

24.  475  dollars. 

57.  789600. 

26.  2300  dollars. 

25.  1591  dollars. 

58.   16170000. 

27.   1372  dollars. 

26.  1950  shil. 

59.  262660000. 

28.  2720  dollars. 

27.  1575  dollars. 

60.  7500  minutes 

29.  4837  dollars. 

28.  2430  shil. 

61.  2400  dollars. 

30.   7785  dollars. 
31.  7744  dollars. 

29.  3936  ounces. 
30.  10754  dollars. 

62.  6800  shillings. 
63.  27000  dollars. 

32.  8820  dollars. 
83.  2  1285  dollars. 

31.  6710  miles. 
32.  8760  hours. 

64.  352500  days. 

136 


ANSWERS. 


[PAGES  47 — 55, 


MULTIPLICATION   CONTINUED. ARTS.  47,  48. 


Ex.     Ana. 

Ex.      Ans.       Ex.       Ana. 

65.  Given. 

78.  2520000.      91.  5816049  galls. 

66.  19500. 

79.  65000000.     92.  101198340  d. 

67.  40800. 

80.  722000000.    93.  146460440  T. 

68.  504000. 

81.  21000000000.  94.  1190439180. 

69.  800000. 

82.  72800000000.  95.  3759670720. 

70.  3300000. 

83.  2240000yds.   96.  4223213600. 

71.  14620000. 

84.  140000  miles.  97.  5815178600. 

72  65360000. 

85.  700000  dolls.   98.  12976172335. 

73  104520000. 

86.  504000  dolls.   99.  124811441568 

74.  183244000. 

87.  27375000  d.  100.  54719418834. 

75.  Given. 

88.  367608  Ibs.   101.  469234745451 

76.  420000. 

89.  3838460ft.   102.  197118900. 

77.  442000. 

90.  4217202  r.    103.  420152303451. 

SHORT  DIVISION. 

ART.  54. 

17.  25. 

9.  116*. 

1.  Given. 

18.  76. 

10.  728.' 

2.  21. 

19.  456. 

11.  1552?. 

3.  23. 

12.  1004f 

4.  122. 

ART.  57. 

13.  400f. 

5.  111. 

20.  Given. 

14.  903f. 

6.  342. 

21.  509. 

15.  923. 

7.  1122. 

22.  901. 

16.  1222f, 

8.  1321. 

23.  1067. 

17.  875. 

9.  1111. 

24.  503. 

18.  1011-|. 

25.  Given. 

19.  63  pair. 

ART.  55. 

20.  42  hats. 

10.  Given. 

ART.  61. 

21.  24  marbles. 

11.  71. 

1.  142. 

22.  45  children. 

12.  43. 

2.  101|. 

23.  75  yards. 

13.  412. 

3.  76. 

24.  85  barrels,  and 

14.  411. 

4.  75. 

5  dolls,  over. 

5.  102f 

25.  92  days. 

ART.  56. 

6.  56f. 

26.  158-J-  yards. 

15.  Given. 

7.  120f. 

27.  195  hours. 

16.  14. 

8.  95. 

28.  333£  hours 

PAGES  56 62. "j  ANSWERS. 


is: 


LONG  DIVISION. 


Ex.              Ans. 

Ex.              Ans. 

Ex.               Ans. 

ART.  62. 

22.  16-ft  shillings. 

5L  89^2^.. 

1,  2.  Given. 

23.   10;HJ-  pounds. 

f\^       1  f  Q    4  7 

3.  128.* 

24.   16|f  pounds. 

53.  218tW 

4.  364. 

25.   17  trunks. 

54.  21&fff. 

5.   1825-f. 

26.  30  weeks. 

6.  533. 

27.  32ft  yards. 

ART.  67. 

7.  732. 

28.  75  dresses. 

55,  56.  Given, 

8.  931. 

29.  81  sheep. 

57.  46-nfv. 

9  —  11.  Given. 

30.  73|f  acres. 

58.  5-iVoV 

31.  61  shares. 

CO        Q    4  6  5 

ART.  65 

32.  3  Iff  years. 

60.'  26^. 

1.  46i. 

•33.  48H  hhds. 

61.   1  S-6-^  6-6g 

2.  48f. 

34.   43ff  months. 

62.  24-1VoioV(F- 

3.  80*. 

35.  5  Iff  months. 

03<   231A5VoW«>- 

4.  40-ft-. 

36.   50  dollars. 

5.  58-ft-. 

37.   10||  months. 

ART.  68. 

6.  48. 

38.   90  pounds. 

C4.   65.  Given. 

7.  21-fV 

31*.  00,  and  1  nv<>r. 

66.   121^£. 

8.  32V?. 

40.   106,  and  22 

67.  32-5^. 

9.  41-ft. 

over.               '68.  54-ifi-4-3{£F. 

10.  27. 

41.  26,    ana    28 

(39^   5]|.a^ 

11.  23-S-f. 

over. 

70.  46^^-8-5J-. 

12.   21|f. 

42.  42,    and    28 

71.  4|-§^-g-. 

13.   Ifjff. 

over. 

72       |  f)    34ft. 

14.   20. 

43.  30|f. 

73.  28-r-Ho-. 

15.   21  ff 

44.  34. 

74.  27dtHf. 

16.   32£f. 

45.  53fV. 

75.  42,AVo. 

17.   45|f. 

46.   35if. 

18.  57|f. 

47.  2$H. 

77.  296Sff^ 

19.  24  caps. 

48.   26|-|. 

78.  17-Ji^-ji-S 

20.  35  pair. 

49.   65-4aff. 

79.  30. 

21.  28  barrels. 

50.   85^-J.    * 

80.  1483-Hi 

81.*  1370iJ-f. 

85.     266^^f. 

89.      194-?^4^4 

82.    19001$  fr. 

86.     184if|^. 

90    139  S-4-^^  A  -ft 

83.      840fff. 

87.  1620-Hi*. 

91.   1171  -J^--5-  Jf  x. 

84.      374-jVVV. 

88.  23664f-f-|. 

138  ANSWERS.  [PAGES  64-74. 

GREATEST  COMMON  DIVISOR.     ART.   74. 


Ex.         Ana. 

Ex.        Ans. 

Ex.        Ans. 

Ex.        Ans. 

2.   28. 
3.  29. 

4.    37. 

5.  79. 

6.  61. 

7.     9. 

8.       8. 

9.     2. 

LEAST  COMMON  MULTIPLE. 


ART.  SO. 

14.  720. 

18.  990. 

24.  8640. 

10.  Given. 

15.  72. 

20.  858. 

25.  13859080. 

11.  80. 

16.  5600. 

21.  7200. 

26.  288000. 

12.  84. 

17.  20160. 

22.  34560. 

27.  3300500. 

13.  180. 

18.  1275. 

23.  1584. 

REDUCTION  OF  FRACTIONS.    ART.  89. 


1.  Given. 

16.  6f. 

31.  YT- 

46.  if. 

2.  Given. 

17.  3}. 

32.   YJT. 

ART.  93- 

3.  f  . 

18.  7i. 

33.    5|2. 

48.  Given. 

4.  f. 

19.   6f. 

34.   3fjl. 

49.  f. 

5-  I- 

20.   1. 

35.   2|^-. 

50.    yL. 

6.  f 

21.  3. 

36.   2ff5. 

51.   1. 

7.  -f. 

22.  4. 

ART.  92. 

52.  T3T. 

O           jj 

23     3  ^  3- 

38.  Given. 

53.  f. 

9!  f* 

24.   5^44- 

39.  /o-. 

54.  JL. 

10.  Tv 

ART.  91. 

40.  Iff. 

55,    -4y4-- 

il-  B- 

26.   Given. 

56.   ^. 

12.   |f. 

27.  4_2. 

42.  |.85' 

57.   1121. 

ART.  90, 

28.   V- 

43.  J-. 

58.   36. 

14.  Given. 

29^    ijL5 

44.   157i 

59.  154. 

15.  3. 

30.   2|4- 

45.  14- 

60.  55. 

ART.  95. 
63-  if,  if,  if- 

64.  Iff,  IB,  iff- 

65.  Iff,  fit,  |*f 

66.  TV 


69. 


HHtfc 


M9340 

TT  oJ  2-^» 


iWiV 


AET.  96. 

71.  Given. 

72.  Given. 

'3.  A,  If-  A- 

74.  f  f,  Jf,  if- 

75.  A,  «,  A- 

76.  f|,  ^|,  ff,  $f. 

77.  f  Hf,  if  if,  f  f  f  f 

*7Q       _7  5  6          735 
" 


PAGES  75-83.] 


ANSWERS. 


139 


REDUCTION  OF  FRACTIONS   CONTINUED. — AW.   96. 


Ex. 


Ans. 


Ex. 


»»•  AV,  W.-AV- 

so,  H,  if,  A,  W- 

.81-  VM- 

82.  H,  W,  W- 

83.  A,  W- 

84.  H.  ft- 

85.  T¥»,  AV,  AV,  AV- 


Iff,  AV- 


88.  ff,  1*. 

89.  T4082,   TVs,  AV- 

90.  AV,  AV.  AV- 

91-  H,  «,  «• 

92.  if,  if,  «. 


ADDITION  OF  FRACTIONS.     ART.  98. 


2.  2|. 

9.  Iff. 

14.  2£i. 

19.  79f 

3.  2f. 

10.  2TVF. 

15.  !TVoV 

20.  334|f. 

4.  If 

H.  lf|; 

16.  8B- 

21.  298TV 

5.  If. 

17.  27T5T. 

22.  505TV 

8.  If  T. 

13!  i|!° 

18.  239ff. 

23.  386if. 

SUBTRACTION  OF  FK  ACTIONS     ART.  100. 


1.  Given. 

2.  A- 

3*     2"V* 

4.  H. 
5-  tVV 
Mf 


9.   ff. 

10.  if. 

H.   A- 
12.    TW 


18.  5|. 

19.  411- 

20.  31 


21-  ^U- 

23.  61. 

24.  8f 

26.  11. 

27.  222V 

28.  31. 


MULTIPLICATION  OF  FRACTIONS.     ART.  102. 


1  Given. 

ART.  1O3. 

24.  14f. 

37.  «. 

2.  3  ^. 

13.   631 

25.   331. 

38.  f. 

3.  8J-. 

14.    1911. 

26.  61T4T. 

39.  fy. 

4    4-4- 

15.   2501. 

27.  48TV 

40.  rVV 

5*.  TTV 

16.   296. 

ART.  1O5. 

41.  if. 

6.  4}f. 

17.  5501. 

29.   80f. 

42.  f-i- 

7.  1324. 

18.   1131. 

30.   211f 

43.  411. 

8.   10§J. 

ART.   1O4. 

31.  554f. 

44.  48. 

9.  I7ff 

20.   12|. 

32.   743f. 

45.   343f 

10.   231. 

21.   lOf 

ART.  1O6. 

46.   21011. 

11.   26!ff. 

22.  24|. 

35.  f. 

47.   3240sV 

12.  Given. 

23.  35f. 

36.  ^ 

48.   5822|i. 

140 


ANSWERS.  [PAGES  84-106. 

DIVISION  OF  FRACTIONS. 


Ex.        Ans. 

Ex.         Ans. 

Ex.        Ans. 

Ex.        Ans. 

ART.  1O8. 

9.     gV 

17.  Given.    - 

26.    6/y. 

1.  Given. 

ART.   1O9. 

18.   1T7F. 

ART.    1J.O. 

2.  Given. 

10.  Given. 

19.   16|. 

27.   Given. 

3.  Given. 

11.  Given. 

20.   7i. 

28.   161. 

4.  T5T. 

12.  Given. 

21     -£~ 

^A«     200' 

29.   168. 

•    1FT* 

13.  li. 

22.  Given. 

30.    196T4T. 

6-  ViV- 

14.  if. 

23.   31. 

31.   484f. 

7.  ^Vj* 

15.  11. 

24.   1411. 

32.   14if 

8«     2V- 

16.  11. 

25.   If. 

33.  1711. 

COMPLEX  FRACTIONS. 


ART.  111. 
1.  Given. 

2.  f. 
3.  |. 

4.  f. 
5.  If- 

0-  *f  ,  or  f 

7.  *-%*. 

EXERCISES   IN  FRACTIONS. 

i.  •*!*• 

2.  45|  m. 

3.  $799TV. 
4.  28641  m. 

5.  $276}i. 
6.  $6.19TV 

7.  $44.37f 
8.  $251. 

9.  $24. 
10.   5?9¥  Ibs. 
11.  10  or. 
12,  5-fj  Ibs. 

13.  16TV?_m. 
14.  21  Ibs. 
15.   1152  Ibs. 

ADDITION  OF  FEDERAL  MONEY.     ART.  117. 

2.^1278.699. 
3.$11261.52. 
4.  $2622.337. 
5.15599.332. 

6.  $1743.828. 
7.82478.735. 

8.  $10224.78. 
9.  $12858.266. 

10.  $978.297. 
11.  $2037.379. 
12.  $880.317. 
13.  $301.243. 

14.  $829.496. 
15.  $34.75. 
16.  $74.375. 
17!  $162.06! 

SUBTRACTION  OF  FEDERAL  MONEY.      ART.  118. 

2.  $468.851. 
3.  $497.73. 
4.  $527.247. 
5.  $5916.707. 

6.  $877.155. 
7.  $461.543. 
8.  $435.103. 
9.  $1461.78. 

10.  $81980.755. 
11.  $67671.133. 
12.  $0.89. 

13.  $2.317. 

14.  $49.928. 
15.  $357.04. 
16.  $2.125. 
17.  $1.945. 

PAGES  107-114.]  ANSWERS.  141 

MULTIPLICATION  OF  FEDERAL  MONEY.    ART.  119. 


Ex.           Ans. 

Ex.           Ans. 

Ex.         Ans. 

Ex.         Ans. 

1.  Given. 
2.  $5070. 
3.  $7250.625. 
4.$21097.80. 
5.1335636.62 
6.  $255991.68 

"7.1458122.602 
8.  $773262.87. 
9.$2182139.52 
10.11.36. 
11.  $10.44. 
12.  $31. 

13.  $78.75. 
14.$12.375. 
15.  $39.45. 
16.  $9.375. 
17.  $23.75. 
18.  $1181.28 

19.  $3346.50 
20.  $1495. 
21.  $4238.08 
22.  $7.50. 
23.873.50. 
24.  $279.50. 

DIVISION  OF  FEDERAL  MONEY.     ART.  120. 


1.  Given. 

7.  $9933.57. 

13.  $8902.627. 

19.  $26.82. 

2.  $142.7  12t. 

8.  $11.322. 

14.  $972.38. 

20.  $8.35. 

3.  $1195.956. 

9.  $110.57. 

15.  $40.69. 

21.  $2.767. 

4.  $806.012. 

10.  $68.47. 

16.  $6.12. 

22.  $1.738. 

5.  $32.16. 

11.  $92.09. 

17.  $7.31. 

23.  $6.807. 

6.  $96.70. 

12.  $49.32. 

18.  $20.16. 

REDUCTION  DESCENDING. 

ART.  124. 

11.  7348  gr. 

20.  24640  r. 

28.  180  pk. 

1-4.  Given. 

13.  212  oz. 

21.  56  qrs. 

2k  52  pts. 

5.  4320d. 

14.  67  Ibs. 

22.  344  na. 

30.  1680  qts. 

6.  469s. 

15.  1728  dr. 

23.  286  ua. 

31.  2520  hrs. 

7.  827  far. 

16.  19696  oz. 

24.  92  qts. 

32.  36000  m. 

8.  Given. 

17.  5120  r. 

25.  976  g. 

33.  954000s. 

9.  156  oz. 

18.  5568  in. 

26.  2016  g. 

34.  524160  m. 

10.  1020  pwt. 

19.  12612  ft. 

27.  10334  q. 

35.  5875200s. 

REDUCTION  ASCENDING. 


ART.  127. 

10.  1    oz.   2  pwts. 

1  5.  2  tons. 

1—4.  Given. 

20  grs. 

16.  24  yds.  1  in. 

5.  27  shillings. 

11.  39  Ibs.  14  oz. 

17.  60  r.  10  ft. 

6.  20  shillings. 

12.  29  qrs.  11  Ibs. 

18.  8  mile*. 

7.  £l,6s.0d.2far. 

13.  1    cwt.   4    Ibs. 

19.   1  in.  6  fur.    32 

8.  22  Ibs.  1  oz. 

11  oz. 

r.  5  yds. 

9=    3  Ibs.  0   oz,    8 

14,  3  Ibs,  14  oz.  8 

20.  2   lea.  1    m.  3 

pwtfi. 

fti 

fu:-.  5  r, 

ANSWERS.          [PAGES  115-118. 


REDUCTION   ASCENDING  AND   DESCENDING. 


Ex.               Ans. 

Ex.               Ans. 

Ex.                Ans. 

ART.  129. 

31.  540  ounces. 

62.  23  gals.  3  qts. 

1.  316  cents. 

32.   1704  scruples 

Ipt. 

2.  812  mills. 

33.  536640  grs. 

63.  3hhds.  48  gls 

3.  2  dolls.  45  cts 

34.   13  Ibs.  9  oz. 

64.  28  gals. 

4.  3  dimes  2  cts 

35.   10   Ibs.    0  oz 

65.  2376  qts. 

1  mill. 

2  drs. 

66.  884  pints. 

5.  95000  cents. 

36.  17  yds.  2  ft. 

67.  4hhds.  40  gls. 

6.   16000  cents. 

37.  46  rods  4  ft. 

68.   1  bbl.  4  gals. 

7.  3170  dimes. 

38.  21120  feet. 

69.   19952  pts. 

8.  4  dolls.  56  cts 

39.  3588  inches. 

70.   12  hhds. 

1  mill. 

40.   1  mile. 

71.   39  bushels. 

9.  8E.2dolls.50c. 

41.  696960  in. 

72.  7    bu.    1    pk. 

10.  61123  mills. 

42.  4  yds.  3  qrs. 

6  qts. 

11.  356  shillings. 

43.  87  qrs. 

73.   22  pks.  4  qts. 

12.  938  farthings. 

44.  568  nails. 

74.  235  pecks. 

13.  £5. 

45.  66Fl.e.  2  qrs. 

75.   762  quarts. 

14.   75s.  6d.  2  far. 

46.  40  E.  e.' 

76.   11  bu.  2  pks. 

15.   18240  far. 

47.  33  F.  e.  2  qrs. 

5  qts. 

16.   60  shillings. 

48.   592  sq.  ft. 

77.  6432  pints. 

17.  5082  pence. 

49.   1194^sq.yds. 

78.  960  minutes. 

18.  £3,    17s    6d. 

50.   H76120sq.ft. 

79.   86400  sec. 

1  far. 

51.   2  A.  25  sq.  r. 

80.  525600  min. 

19.   84  ounces. 

52.   15    sq.    r.    7f 

81.  4  days. 

20.  2200  pwts. 

sq.  yds. 

82.  2  days  12  hrs. 

21.   13  Ibs.  9  oz. 

53.   6    sq.    ft.     12 

56  min. 

22.   1  oz.  15  pwts. 

sq.  in. 

83.   604800  sec. 

23.   19735  grains. 

54.   1296  cu.  ft. 

84.   8  yrs.  11  mo. 

24.   1   Ib.  2  oz.    5 

55.  93312  cu.  in. 

85.   1410'. 

pwts.  20  grs. 

56.   3328  cu.  ft. 

86.   147600". 

25.    12  Ibs.  8  oz. 

57.  2  cu.   ft.  774 

87.   14°. 

26.  4176  ounces. 

cu.  in. 

88.   Os.  16°  4  . 

27.  6200  Ibs. 

58.   25    cords,   64 

89.   216000". 

28.  2400  ounces. 

cu.  ft. 

90.   9120'. 

29.  62  Ibs.  8  oz. 

59.  756  pts. 

91.  0°  16'. 

30.  2  tons,  2  cwt. 

60.  2200  gills. 

92.   1  sign. 

2  qrs.  6  Ibs. 

01.  2580  qts. 

PAGES  119-125.]  ANSWERS.  143 

REDUCTION  ASCENDING  AND  DESCENDING. 


Ex.      Ans. 

Ex.      Ans. 

Ex.      Ans. 

93.  45360  far. 
94.  £63. 
95.  248G.5s.8d. 
96.  80  G. 

104.  13  r.  13  f.  8  i. 
105.  69840  oz. 
106.  205554  grs. 
107.  36  E.  ells. 

116.  194  p.  Ih.  43 
gals. 
117.  473353920s, 
118.  13  wks.  1  d. 

97.  268440  grs. 
98.  143  1.  4  o.  1 

108.  56  yds.  1  qr. 
109.  72.  Fl.  ells. 

22  hrs.  20  min. 
119.  3581  793s.  ft 

p.6g. 
99.  357360  grs. 
100.  33109  Ibs. 

110.  40  F.  ells. 
111.  839599  in. 
112.  5  1.  306  r.  7  f. 

120.  39  ft.  1208  i, 
121.  2222208  c.in. 
122.  17003520  in. 

101.  24  T.  9  cwt. 
10  Ibs.  8  oz, 
102.  3682  in. 
103.  58278  ft. 

113.  8693  pts. 
114.  165qrs.  2  bu. 
2  pks.  5  qts. 
115.  97344  gills. 

123.  13  A.  75  r. 
1  1^  yds. 
124.  1018818  sec. 
125.  41  S.24°,20'. 

COMPOUND  ADDITION. 


5.  £40,  14s.2d.  2  f  . 

9.  35  w.  4h.  21m. 

13.  6  pi.   18  gals. 

6.  59  1.  2  p.  22  g. 

10.  23  yds.  3  na. 

3  qts.  2  gi. 

7.  22  r.  1  yd.  5  in. 

11.  13  T.  12  c.l  qr. 

14.  37  bu.  3  pks. 

8.  26  cwt.  3  qrs.  5 

10  1.  9  o.  13  d. 

3  qts. 

Ibs.  5  oz. 

12.  27hhds.38g. 

15.  67  y.  3q.  2  na. 

COMPOUND  SUBTRACTION. 

2.  £7,  Is.  9d.  2  far. 

6.  7  yds.  3  qrs.  1  n. 

12.  4  A.  2  roods,  4 

3.  3   Ibs.   7   oz.   4 

7.  £9,  17s.  4d. 

rods. 

pwts.  8  grs. 

8.  44  gals.  1  pt. 

13.  8  lea.  2  mi.  6 

4.  4  T.  17  cwt.  1 

9.  4T.  16c.741b. 

fur.  4  r. 

qr.  24  Ibs. 

10.  2y.  3  mo.  16d. 

14.   17  bu.  5  q.  2  p. 

5.  5  m.  5  fur.  7  r. 

11.   15  y.  10  mo.  3 

15.  45  G.  18s.  8d 

3  ft.  9  in. 

d.  8  h.  4  in. 

2  far. 

COMPOUND  MULTIPLICATION. 

1.  Given. 

5.  661.  285  r.  11  f. 

9.  625  y.  2  q.  2  n. 

2.  £64,   13s.  5d.  1 

3  i. 

10.  2173  d.  13  h.  3 

farthing. 

6.  199  T.  14  c.  14 

in.  47  s. 

3.  236  1.5  o.  11  p. 

1.  15  o.  1  d. 

11.  2272   y.   30  w 

12  g. 

7.  43A.16r.84ff. 

3  d.  12  h. 

4.  34  mi.  2  f.  20  r. 

8.  10  cords,  61  c.  f. 

12.  4707  h.  18  g. 

144 


ANSWERS.  [PAGES  125-132. 

COMPOUND  DIVISION. 


Ex.               Ana. 

Ex.               Ans. 

Ex.              Ans. 

3  .  Given, 
2.  4  1.  9  oz.   15   p. 
11  g. 
3.  £1,  9s.  5d.  OJ  f. 
4.  5  1.  13  p.  15£  g. 
5.  2  T.   12   c.  2  q. 
U|  1. 

6.  3  y.  1  q.  If  na. 
7.  4    L    1   m.  2  f. 

174* 

8.  5h.2g.4q.0£p. 
9.  3  b.  3  p.  3f  q. 
10.  £5,  3s.  6d.  2  f. 
11.2  T.  5  c.  3|il. 

12.  1  hhd.  49  gals. 
3ff  qte. 
13.  1    A.  2   roods, 

ll-ft* 

14.  91  c.  f.  OH  i. 
15.  1   yr.    12  wks. 
4ffr<L 

MISCELLANEOUS  EXERCISES. 

1.  271557. 

23.  460^. 

51.  $279. 

76.  150  ap. 

2.  1229930. 

24.  17-&V 

52.  $1180. 

77.  112  mar. 

3.  875431. 

25.   IS-ftV. 

53.  3399  s. 

78.  192  par. 

4.  372713- 

26.   12|«f 

54.  $90. 

79.  371  fans. 

124. 

27.  43+-Hft. 

55.  87  s. 

80.   25  2f  yds. 

5.  837598- 

28.   74-AVgV 

56.   13-ft-  yds. 

81.  512  pair. 

35. 

29.  261. 

57.  13-/5-  yds. 

82.   10962d. 

6.  2406iH- 

30.  508280. 

58.  33-^  a. 

83.  60850  f. 

7.   1593i*f 

33.  $19. 

59.  20  cows. 

84.  302  yds. 

8.  245948, 

34.  $13. 

60.  25  bar. 

85.  4000. 

616  rein. 

35.  $17. 

61.  29  or. 

86.   1890  s. 

9.  41* 

36.  $49. 

62.  64  quarts. 

87.  $20. 

10.  6800. 

37.  86  Ibs. 

63.  60  Ibs. 

88.  7560  bot 

11.  98040. 

38.  63  s. 

64.  75  c. 

89.  $40.80. 

12.  53f£. 

39.  356  Ibs. 

65.  $108. 

90.  $124.16. 

13.   108332. 

40.  $28. 

66.  $7.92. 

91.  $1100. 

14.   11542. 

41.   300  m. 

67.  $252. 

92,  $442. 

15.  33611. 

42.  $19. 

68.  936  in. 

93.  $3180. 

16.  268G9. 

43.  $18. 

69.  45  cents. 

94.  $9600. 

17.  3810225. 

44.  $75. 

70.  $30. 

95.  $17280. 

18.  12469. 

45.   106  sh. 

71.  $140. 

96.  $1981980. 

19.  2720325. 

46-.  $837. 

72.  $150. 

97.  $960. 

20.   17. 

47.  $2008. 

73.  12  dress's. 

98.  $130.56. 

21.  5  and  9  r. 

48.  $1744. 

74.  16  boxes. 

99.  52  y.  6  m. 

22.  18  and 

49.  $114. 

75.  110  miles. 

100.  4  weeks. 

424  over. 

50.  $165. 

APPENDIX. 


METRIC   WEIGHTS  AND  MEASURES. 

ART.  1 .  The  Metric  System  of  Weights  and  Measures 
is  founded  upon  the  decimal  notation,  and  is  so  called 
because  its  primary  unit  or  base  is  the  Meter. 

2.  The   meter   is    the    unit  of  length,   and    is    equal 
to  one  ten-millionth  part  of  the  distance  on  the  earth's 
surface  from  the   equator   to  the    pole,  or   39.37  inches 
nearly. 

3.  From    the  meter    are  derived  the  unit  of  surface 
called  the  are,  the  unit  of  capacity  called  the  liter,  and  the 
unit  of  weight  called  the  gram. 

4,9  The  several  ascending  and  descending  denomina- 
tions increase  and  decrease  regularly  by  the  scale  of  ten, 
according  to  the  law  of  simple  numbers.  (Art.  9.) 

»>.  Tlie  names  of  the  higher  denominations  are  formed 
by  prefixing  to  the  several  units  the  Greek  numerals,  dec' a, 
hec'to,  kil'o,  and  myr'ia,  which  denote  10,  100,  1000, 
10000 ;  as  dec'ameter,  hec'tometcr,  kilfometer,  myr'ia- 
meter.  ^ 

The  names  of  the  lower  denominations  are  formed  by 
prefixing  to  the  units  the  Latin  numerals,  dcc'i,  cen'ti,  and 

QUEST. — 1.  Upon  what  is  the  Metric  System  founded?  Why  so 
called?  2.  What  is  the  meter?  3.  From  what  are  the  units  of  sur- 
face, capacity,  and  weight  derived  ?  4.  How  do  the  ascending  and 
descending  denominations  increase  ?  5.  How  are  the  names  of  th« 
higher  denominations  formed  ?  The  lower  ? 


2  METRIC    SYSTEM. 

mil'li,  which  denote  j\,  T^,  and  j^^  ;  as  dec'imeter,  cen  - 
tirneter,  and  miriimeter. 

NOTE. — The  numeral  prefixes  are  the  key  to  the  system,  and 
therefore  should  be  thoroughly  learned  at  the  outset. 

LINEAR   MEASURE. 

O.  The  unit  of  length  is  the  meter.  The  denomina- 
tions are  the  mil'limetcr,  cen' timeter,  decimeter,  me'ter,  dec'a- 
meter,  hec' tomcter,  kiUometer^  and  myr'iameter. 

Denominations  Equivalents.* 

10  imTli-me-ters  (mm.)  =  1  cen'ti-me-ter,    (cm.)  =  .3937  in. 

10  cen'ti-me-ters  =  1  dec  i-mo-ler,     (dm.)  =  3.937  in. 

10  dec'i-me-ters  =  1  mo'cer  (m.)     =  39.37  in. 

10  me'ters  =  1  doc'a-me-ter,  (Dm.)  =  393.7  in. 

10  dec'a-me-ters  =  1  hec'to-me-ter,  (Hm.)  =  328    ft.    1    in. 

10  hec'to-me-ters  =  1  kil'o-me-ter,    (Kin.)  =  3280  ft.  10  in. 

10  kil'o-me-ters  =  1  myr'i -a-me-ter,(Mm.)=  6.2137     miles. 

Approximate  values,  expressed  in  round  numbers,  are  often  use- 
ful in  comparing  Metric  Weights  and  Measures  with  those  in  com- 
mon use.  The  following  are  proposed: — 

Consider  a  meter  40  inches ;  a  decimeter  4  inches ;  5  meters  1  rod ; 
a  kilometer  200  rods,  or  £  of  a  mile,  &c. 

NOTES. — 1.  The  term  meter,  is  from  the  Greek  metron,  a  measure. 
The  standard  meter  is  a  scale  made  of  platinum,  and  is  preserved 
in  the  National  Archives. 

2.  The  denominations  most  used  in  linear  measure,  are  the  centi- 
meter, meter,  and  kilometer.  Long  distances,  as  roads,  canals,  &c., 
are  reckoned  in  kilometers;  short  distances,  as  cloths,  ribbons, 
&c:,  are  estimated  by  the  meter  and  centimeter.  The  millimeter, 
decimeter,  decameter,  &c.,  like  mills,  dimes,  and  eagles,  in  Federal 
money,  are  seldom  used. 

In  reciting  the  tables  the  last  column  may  be  omitted. 

QUKST. — 6.  "What  are  the  denominations  in  linear  measure  ?  Re- 
peat the  table.  Note.  "What  are  the  denominations  most  used  in 
this  measure? 

*  Act  of  Congress  1 866. 


METRIC    SYSTEM.  3 

SQUARE   MEASURE. 

7.  The  square  meter  is  the  unit  for  measuring  ordinary 
surfaces,  as  floors,  ceilings,  &c.     For  smaller  surfaces,  the 
square  decimeter,  centimeter,  &c.,  are  employed.* 

As  the  meter  contains  10  decimeters,  a  square  meter 
must  contain  100  square  decimeters,  for  10x10  =  100. 
For  the  same  reason  a  square  decimeter  must  contain  100 
square  centimeters,  &c. 

Denominations.  '     Equivalents.f 

100  sq.  mirii-me-ters  =  1  sq.  ceu'ti-me-ter,     =     0.155  sq.  in. 

100  sq.  cen'ti-me-ters  =  1  sq.  dec'i-me-ter,        =     15.5     sq.  in. 

100  sq.  dec'i-me-ters  =  1  sq.  me'ter,                —     1550    sq.  in. 

Approximate  values. — Consider  a  sq.  meter  10  sq.  ft. :  a  sq.  deci- 
meter, 15  sq.  in.,  &c. 

NOTE. — Since  square  decimeters  are  liundredths  of  ^  square  meter ; 
square  centimeters  are  hundredth^  of  a  square  decimeter,  &c., — it  fol- 
lows, that,  in  writing  them,  sq.  decimeters,  sq.  centimeters,  &c.,  must 
each  occupy  two  decimal  places.  Hence,  if  the  number  of  sq.  deci- 
meters be  less  than  10,  a  cipher  must  be  prefixed  to  the  figure 
denoting  them.  Thus,  5  sq.  meters  and  7  sq.  decimeters  are 
written  5.07  sq.  meters;  87  sq.  meters,  3  sq.  decimeters  and  9 
sq.  centimeters  are  written  37.0309  sq.  meters. 

8.  The  unit  for  Land  Measure  is  called  the  are,  which 
is  equal  to  a  sq.  decimeter,  and  therefore  contains  100  sq. 
meters.      The  denominations  of   Land    Measure    are  the 
cen'ti-are,  the  are,  and  the  liect'are* 

Denominations.  Equivalents.f 

100  cen'ti-ares  (ca.)        —        1  are,  (a.)  =     119.6  sq.  yards. 

100  ares  —        1  hect'are,  (Ha.)  =     2.471  acres. 

QUEST. — 7.  "What  is  the  unit  for  measuring  ordinary  surfaces?- 
8.  "What  in  measuring  land?  Note.  Why  are  there  no  deciare  and 
decare  in  land  measure? 

*  LamoUe.  f  Act  of  Congress,  1866. 


4  METRIC    SYSTEM. 

Approximate  values. — Call  a  centiare  10  sq.  ft.;  an  are  4  sq  rods, 
a  hectare  2£  acres. 

NOTES. — 1.  The  term  are,  is  from  the  Latin  area,  a  surface. 
Small  pieces  of  land,  as  grass  plats,  court  yards,  &c.,  are  commonly 
estimated  by  square  meters. 

2.  It  will  be  observed  that  there  is  no  declare  (^  of  an  are)  nor 
decare  (10  ares).  The  reason  is  that  the  decimal  scale  is  applied  to 
the  length  of  the  sides  of  the  squares,  instead  of  their  surfaces. 
Thus,  the  sides  of  the  centiare,  the  are,  and  the  hectare^  are  respec- 
tively 1  meter,  1  decameter,  and  1  hectometer  in  length ;  and  their 
surfaces  are  1  sq.  meter,  100  sq.  meters,  and  1000  sq.  meters. 

Had  the  decimal  scale  been  applied  to  the  surface,  the  sides  of 
surfaces  containing  10  square  meters,  1000  square  meters,  &c., 
could  not  be  expressed  with  exactness  in  decimals,  and  to  obtain 
them  it  would  be  neccessary  to  extract  the  square  root. 

CUBIC    MEASURE. 

O.  The  unit  for  measuring  ordinary  solids,  as  embank- 
ments, excavations,  &c.,  is  the  cubic  meter.  Smaller  bodies 
are  estimated  in  cubic  decimeters,  centimeters,  or  milli- 
meters. 

Since  each  side  of  a  cubic  meter  is  10  decimeters  in 
lerrgth,  it  follows  that  a  cubic  meter  must  contain  1000 
cubic  decimeters;  for  10  x  10  X  10  =1000.  Also,  that  a 
cubic  decimeter  contains  1000  cubic  centimeters,  &c. 

Denominations.  Equivalents.* 

1000  cu.  mil'limeters  =  1  cu.  cen'timeter,  =  0.061  cu.  in. 
1000  cu.  cen'timetera  =  1  cu.  dec'imeter,  —  61.026  cu.  in. 
1000  cu.  dec'imeters  =  1  cu.  me'ter,  =  35.316  cu.  ft 

NOTE. — As  cubic  decimeters  are  thousandths  of  a  cubic  meter, 
cubic  centimeters  thousandths  of  a  cubic  decimeter,  <fcc.,  it  follows 
that  cubic  decimeters,  centimeters,  &c.,  must  each  occupy  three  decimal 
places;  consequently,  if  the  number  of  cubic  decimeters,  <fec.,  is 

QUEST.— 9.  What  is  the  unit  for  measuring  ordinary  sol'ds? 
Note.  How  many  places  do  cu.  decimeters  occupy  ? 

*  Act  of  Congress,  1866. 


METRIC    SYSTEM.  5 

less  than  100,  ciphers  must  be  prefixed  to  the  figure  or  figures 
denoting  them.  Thus,  73  cubic  meters  and  5  cubic  decimeters  are 
written  73.005  cubic  meters. 

1O.  The  unit  for  measuring  wood  and  timber  is  called 
the  stere,  which  is  equal  to  a  cubic  meter.  The  stere  has 
only  one  subdivision,  which  is  called  the  dec'i-stere,  and 
one  multiple,  called  the  dec'a-stere. 

Denominations.  Equivalents.* 

10  dec'i-steres  =  1  stere  —  35.316  cu.  ft.,  or  1.308  cu.  yds. 

1Q.  steres  =   1  dec'a-stero     =         2.759  cords,  or  353.6  cu.  ft. 

Approximate  values. — Call  a  (Jecistere  3J-  cubic  feet;  a  stere  J 
cord;  a  decastere  2J-  cords. 

NOTES. — 1.  The  term  stere,  is  from  the  Greek  sterosr  solid. 

2.  In  France,  fire-wood  is   commonly  measured  in  a  cubical  box 
or  crib,  whose  length,  breadth,  and  height  are  each  1  meter. 

3.  In   computing   large   quantities   of  wood,  it  is   customary  to 
reckon  by  steres  or  decasteres. 

DRY  AND  LIQUID  MEASURE. 
11«  The  unit  for  measuring  liquids  and  dry  articles 
as  oil,  wine,  grain,  fruit,  &c.,  is  the  cubic  decimeter,  which 
is  called  the  I'i'ter.  The  liter  has  the  form  of  a  cylinder, 
and  is  equal  to  1.0567  wine  quarts.  The  denominations 
are  the  miVlil'iter,  centiliter,  decfiliter,li'ter,  dec'al'iter, 
hectoliter  and  kil'oliter. 

Denominations.  Dry  Measure.*        Liquid  Measure.* 

10  mil'li-li-ters    —  1  cen'ti-K-ter  =  0.6102  cu.  in.  =  0.338  fid.  oz 

10  cen'ti-li-ters  =  i  dcc'i-li-ter  —  6.1022  cu.  in.  =  0.845        gill 

10  dec'i-Ii-ters    =.-  1  li'ter  =  0.908    quart  —  1.0567  quart 

10  li'ters  =  1  dec'a-li-ter  =  9.08          "  =  2.6417   galls. 

10  dec'a-li-ters   =  1  hec'to-li-ter  =  2.8375  bush.  —  26.417      " 

10  hec'to-li-ters  =  I  kil'o-li-ter  =  28.372      "  —  264.18      " 


QUEST. —  10.  What  is  the  unit  for  measuring  wood?  Name  the 
denominations?  11.  What  is  the  unit  of  dry  and  liquid  measure? 
The  denominations?  Note.  What  denominations  are  used  most? 

*  Act  of  Congress,  1866. 


6 


METRIC    SYSTEM. 


Approximate  values. — Call  a  liter  1  quart,  and  a  hectoliter  2$ 
bushels. 

NOTES. — 1.  The  term  liter  is  from  the  Greek  litron,  a  measure  of 
capacity. 

2.  The  denominations   chiefly  used   in   liquid   measure,  are    the 
liter t  decaliter,  and  deciliter;  in  dry  measure  the  liter,  decaliter,  hec- 
toliter, and  kiloliter. 

3.  Since  the  liter  is  equal  to  a  cubic  decimeter,  it  follows  that  the 
kiloliter  (1000  liters)  contains  a  cubic  meter :    that  the  deciliter  (the 
10th  of  a  liter)  contains  100  cubic  centimeters,  &c. 

4.  A  milliliter  of  water  weighs  1  gram ;   a  liter  1  kilogram ;   and 
a  kiloliter,  or  cubic  meter,  1  tonneau,  or  ton. 

WEIGHTS. 

13.  The  unit  of  weight  is  called  the  gram,  which  is 
equal  to  the  weight  of  a  cubic  centimeter  of  distilled  water 
in  a  vacuum,  at  the  temperature  of  39.83°  Fahrenheit,*  or 
15.432  grains.  The  denominations  are  the  mil'li-gram, 
cen'ti-gram,  dec'i-gram,  gram,  dec'a-gram,  hec'to-gram,  kilf- 
o-grani,  myr'i-a-gram,  quin'taL  and  mil' Her  or  ton.\ 


Denominations. 

Equival« 

mts.t 

10 

milligrams         — 

1 

cen'tigram        — 

0.1543 

grains. 

10 

cen'tigrams        = 

1 

dec'igram         = 

1.5432 

u 

10 

dec'igmms        = 

1 

gram                — 

15.432 

" 

10 

grama                 = 

1 

dec'agram        = 

0.3527 

oz. 

avoir. 

10 

dec'agranis        = 

1 

hec'togram       = 

3.5274 

" 

" 

10 

hec'togratns       = 

1 

kil'ogram          = 

2.2046 

Ibs. 

" 

10 

kilograms          = 

1 

myr'iagram      = 

22.046 

" 

" 

10 

myr'iagrama     = 

1 

quin'tal             = 

220.46 

" 

w 

10 

quin'tals 

1 

mil'lier  or  ton  = 

2204.6 

«i 

u 

QUEST. — 12.   What  is  the  unit  of  weight?     Name  the  denomina- 
tions.     % 

*  This  temperature  is  equivalent  to  4*  of  the  Centigrade  Ther- 
mometer, and  is  the  point  at  which  water  attains  its  maximum  density. 
|  Contraction  of  touneau. 
\  Act  of  Congress,  1866. 


METRIC    SYSTEM.  7 

Approximate  values. — Call  a  gram  15  grains  ;  a  kilogram  2£ 
.pounds;  a  quintal  220  pounds,  and  a  tonneau  2200,  or  a  long  ton. 

NOTES. — 1.  The  term  gramis  from  the  Greek  gramma,  a  standard. 

2.  The  denominations  of  weight  most  in  use,  are  the  gram  and 
kilogram*  The  gram  with  its  subdivisions  is  used  in  mixing  medi- 
cines, and  other  cases  requiring  minuteness  and  accuracy.  The 
kilogram,  somtimes  contracted  to  kilo,  is  the  ordinary  weight  of  com- 
merce. In  weighing  heavy  articles  the  quintal  and  ton  are  used. 

FRENCH  CURRENCY. 

13.  The  currency  of  France,  like  the  weights  and 
measures,  is  based  upon  the  decimal  system.  The  de- 
nominations are  the/rawc,  fche  dec'lme,  and  cen't'ime. 

Denominations.  Equivalents. 

10  centimes  =  1  dec'ime  =  0.0186        dollar. 

10  dec'imes  1  franc  =  0.186  " 

NOTE. — The  franc  is  the  unit.  The  dedme,  like  our  dime,  is  seldom 
used ;  its  value  being  expressed  in  centimes  or  hundredths  of  a  franc. 
Thus,  85  francs,  4  decimes,  and  3  centimes,  are  written  85-43  franca. 
Centimes,  being  hundredths  of  a  franc,  require  two  decimal  places. 

14L.  The  coins  of  France  are  of  three  kinds, — gold, 
silver,  and  bronze. 

The  gold  coins  are  40  franc,  20  fratic,  and  5  franc  pieces. 

The  silver  coins  are  the  fra*nc,  2  franc,  and  5  franc  pieces. 

The  bronze  coins  are  1  centime,  2  centimes,  5  centimes, 
and  10  centimes;  which  weigh,  1,  2,  5,  and  10  grains,  re- 
spectively. 

NOTE. — The  standard  of  the  gold  and  silver  coins  is  $>  pure 
inetal,  and  ^o  alloy. 

1.  Write  125  francs  and  7  centimes. 

Ans.  125.07  francs 

2.  Write  260  francs  and  4  decimes. 

3.  Write  907  francs,  3  decimes,  8  centimes. 

*  The  standard  kilogram  adopted  a  as  model  for  weights,  is  made 
of  platinum,  and  preserved  in  the  archives  of  the  government. 


8  METRIC    SYSTEM. 

METRIC  NOTATION. 

JL5»  Ex.  1.  Write  7  kilometers,  5  hectometers,  4  in e^ 
ters,  2  decimeters,,  and  8  centimeters,  in  meters. 

Analysis. — Since  the  denominations 
of  the  Metric  System  increase  and  de-  ™ 

crease  by  the  decimal  scale,  it  is  plain     7504.28  meters, 
they  may  be  written  one  after  another 
like  simple  numbers,  placing  a  decimal  point  between  the 
denomination  regarded  as  the  unit,  and  those  below  it,  to 
show  that  the  latter  are  decimal  parts.     There  being  no 
decameters,  a  cipher  is  put  in  its  place.     The  above  dis- 
tance is  therefore  equivalent  to  7504.28  meters.     Hence, 

1G.  To  express  Metric  Weights  and  Measures. 

Write  the  given  denominations  one  after  another ',  begin- 
ning with  the  highest,  and  place  a  decimal  point  between 
the  one  taken  as  the  unit,  and  those  below  it. 

NOTES. — 1.  If  any  intervening  denominations  are  omitted  in  the 
given  number,  their  places  must  be  supplied  by  ciphers.  (Art.  15.) 

2.  In  Metric  as  well  as  in  Compound  Numbers,  convenience  re- 
quires that  the  measure  employed  as  the  unit,  should  be  proportionate 
to  the  thing  measured.  Thus,  long  distances,  as  from  New  York  to 
San  Francisco,  should  not  be  stated  in  meters,  for  the  reason  that 
the  number  would-be  too  large.  Nor  should  the  meter  be  employed 
to  measure  the  thickness  of  paper,  because  its  thickness  is  too  sriiall 
a  part  of  that  unit. 

1.  Express  5  kiloliters,  8  hectoliters,  7  liters,  and  4  cen- 
tiliters in  hectoliters,  in  liters,  and  centiliters,  respectively. 

Ans.  58.0704  hectoliters  ;  5807.04  liters;  580704  cen- 
tiliters. 

2.  Write    13    quintals,  4    myriagrams,   1    kilogram,  5 
grams,  and  25  centigrams,  making  the  kilogram  the  unit. 

QUEST. — 16.  How  write  metric  weights  and  measures  ?  Note.  If 
any  denomination  is  ommitted.  what  is  to  be  done  ? 


METRIC    SYSTEM.  9 

3.  Write  18  sq.  meters  and  5  sq.  decimeters. 

Ans.  18.05  sq.  m. 

4.  Write  17  hectares,  6  ares,  arid  3  centiares,  in  ares. 

Ans.  1706.03  ares. 

5.  Express  in  cubic  meters,  19  cubic  meters  and  17  cu- 
bic decimeters.  Ans.  19.017  cu.  m 

6.  Express  in  liters,  61  hectoliters,  7  liters,  3  centiliters 
and  5  milliliters. 

REDUCTION  OF  METRIC   WEIGHTS  AND 

MEASURES. 

CASE  I. 

IT*.  To  reduce  Metric  Weights  and  Measures  from  a 
higher  denomination  to  a  lower. 

Ex.  1.  Reduce  46.3275  kilometers  to  meters. 

Analysis. — Since  a  unit 

of  a  higher  denomination  Operation. 

equals  ten  in  the  next  lower,    46.3275  Km.  =  46327.5m. 
it  is  plain,  to  reduce  a  higher 

denomination  to  the  next  lower,  we  must  multiply  by  10  ; 
to  reduce  it  to  the  next  lower  still,  we  must  multiply  it  again 
by  10,  and  so  on. 

But  to  multiply  by  10,  we  remove  the  decimal  point 
one  place  to  the  right,  &c.  (Art.  192.)  Hence,  the 

RULE. — Remove  the  decimal  point  one  place  to  the  right 
for  each  lower  denomination  to  which  the  given  number  is 
to  be  reduced,  annexing  ciphers  if  necessary. 

NOTE. — If  the  given  number  has  no  decimal  figures,  the  decimal 
point  is  supposed  to  occupy  the  first  place  on  its  right. 

QUEST. — 17.  How  reduce  metric  numbers  from  higher  to  lower 
denominations? 


10  METRIC     SYSTEM. 

2.  Reduce  867  kilograms  to  grams. 

Ans.  867000  grains, 

3.  Reduce  264.42  hectoliters  to  centiliters. 

4.  In  2561  ares,  how  many  square  meters? 

5.  In  8652  cubic  meters,  how  many  cubic  decimeters  ? 

6.  In  63240  cubic  decimeters,  how  many  cubic  centi- 
meters. Ans.  63240000  cu.  Cm 

7.  Reduce  4256.25  kilograms  to  grams, 

8.  Reduce  845  francs  to  centimes. 

CASE.    II. 

18.  To  reduce  Metric  Weights  and  Measures  from  a 
lower  denomination  to  a  higher. 

9.  Reduce  84526.3  meters  to  kilometers. 
Analysis. — Since  it  takes 

ten  of  each  lower  denomina-  Operation. 

tion  to  make  a  unit  in  the 

next  higher,  it  follows  that,       84526.3  M.  =  84.5263Km. 

to  reduce  a  number  from  a 

lower  to  the  next  higher  denomination,  it  must  be  divided 

by  10;  to  reduce  it  to  the  next  higher  still,  it  must  be 

divided  again  by  10,  and  so  on.     (Art.  9.)     But  to  divide 

by  10,  we  remove  the  decimal  point  one  place  to  the  left. 

&c.     (Art.  195.)     Hence  the 

RULE. — Remove  the  decimal  point  one  place  to  the  left 
for  each  higher  denomination  to  which  the  number  is  to  bt 
reduced,  prefixing  ciphers  if  necessary. 

10.  Reduce  87  meters  to  kilometers. 

Ans.  0.087  Km. 

11.  In  1482.35  grams,  how  many  kilograms  ? 

QUEST. — 18.  How  reduce  metric  weights  and  measures  from 
lower  to  higher  denominations? 


METRIC     SYSTEM.  11 

12.  In  39267.5  liters,  how  many  kiloliters  ? 

13.  Reduce  812067  centiares  to  hectares. 

14.  In   1000000   cubic  centimeters,  how  many    cubic 
meters  ?  Ans.  1  cu.  m. 

15.  In  605349  cubic  meters,  how  many  kiloliters? 

CASE  III. 

19.  To  reduce  Metric  Weights  and  Measures  to  the 
common  system. 

1.  Reduce   3  hectometers,  6  decameters,   and    5    deci- 
meters to  inches. 

Analysis. — 3  hectometers,  6  Operation. 
decameters,  and   5    decimeters 

—  3 6 0.5  meters.     Now  as  there  360.5       meters, 

are  39.37  inches  to  every  me-  39.37 

ter,  there  must  be  39.37  times  14192  885  in 
as  many  inches  as  meters;   and 
360.5  x  39.37  =  14192.885  in.  or  1182.74  +ft.     Hence  the 

RULE. — Express  the  given  metric  number  decimally  in 
the  denomination  of  the  principal  unit  of  the  table,  and 
multiply  it  by  the  value  of  that  unit ;  the  product  will  bet 
the  answer. 

NOTES. — 1.  The  product  will  be  in  the  same  denomination  as 
that  in  which  the  value  of  the  principal  unit  of  the  table  is  ex- 
pressed, and  may  be  reduced  to  any  other  denomination  required. 
(Arts.  161,  162.) 

2,  The  principal  unit  of  dry  and  liquid  measure  is  the  liter ;  that 
of  weight  is  the  gram   or  kilogram,  &c. 

2.  Reduce  573  kilograms  to  pounds. 

Ans.    1263.2358  pounds 

QUEST. — 19.  How  reduce  metric  weights  and  measures,  to  the 
common  system  ? 


12  METRIC     SYSTEM. 

3.  In  1285  liters,  how  many  wine  gallons? 

4.  In  391  kiloliters,  bow  many  bushels? 

5.  Reduce  1865  meters   £nd  25  centimeters  of  cloth  to 
yards.  Ans.  2039.61206  -h  yards. 

6.  In  35260  ares  of  land,  how  many  acres  ? 

7.  Reduce  508.85  francs  to  Federal  money. 

CASE   IV. 

SO.    To  reduce  common  weights  and  measures  to  the 
Metric  System. 

8.  In  48  rods,  6  feet,  5  inches,  how  many  meters? 

Analysis. — 48  rods,  6  feet,  5 
inches  =  9581  inches.  Operation. 

Now  as  39.37  inches  make  1  39.37  )  9581.00000. 
meter,  9581  inches  will  make  as  040  057  , 
many  meters  as  39.37  is  con- 
tained times  in  9581,  which  is  243.357  in.     Hence  the 

RULE. — Reduce  the  compound  number  to  units  and  deci- 
mals of  a  unit  of  the  same  denomination  as  that  in  which 
the  principal  metric  unit  of  the  table  is  expressed,  and  di- 
*vide  it  by  the  value  of  this  unit ;  the  quotient  will  be  the 
answer. 

NOTE. — The  quotient  will  be  in  the  denomination  of  the  principal 
unit  of  the  table,  whose  value  has  been  employed  as  a  divisor. 

9.  In  3  cwt.  15  Ibs.  12  oz.,  how  many  kilograms? 
Solution. — 3  cwt.  15  Ibs.  12  oz.    =    315.75   Ibs.,  and 

315.75-^-2.0246  — 156.944+ kilograms. 

10.  Reduce  1917  miles  to  the  metric  system. 

Ans.   1191.160  +  Km. 

QUEST. — 20.  How  reduce  dbmmon  weights  and  measures  to  the 
metric  system  ?     Note.  Of  what  denomination  is  the  quotient  ? 


METRIC    SYSTEM.  13 

11.  In  13750  pounds,  how  many  kilograms? 

12.  Reduce  2056  bushels,  3  pecks  to  kiloliters. 

13.  Reduce  9256  sq.  rods  to  ares. 

14.  Reduce  14506  cu.  feet  to  cu.  meters. 

15.  Reduce  $357.375  to  francs. 


ADDITION,    SUBTRACTION,  MULTIPLICATION, 

AND  DIVISION  OF  METRIC  WEIGHTS 

AND  MEASURES. 

2  1  .  Since  the  denominations  of  Metric  Weights  and 
Measures  increase  and  decrease  by  the  scale  of  ten,  those 
above  units  occupying  the  place  of  tens,  hundreds,  thou- 
sands, &c.,  those  below,  tenths,  liundredihs,  <kc.,  it  is  plain 
they  may  be  added,  subtracted,  multiplied,  and  divided,  by 
the  corresponding  rules  of  Decimal  Fractions. 


Ex.  1.  What    is  the    sum   of   7358.356    meters, 
86.142  decameters,  95  centimeters,  and  450.6  meters. 

Analysis.  —  Reducing     the    given  Operation. 

numbers  to  the  same  denominations,      /35S.356       meters. 
viz.,   to    meters,  86.142   decameters       861.42  " 

become     861.42    meters,    and     95  0.095  " 

centimeters   become   0.095    meters.       450.6  " 

We     now     write      the     numbers     8670  471  " 

units     under     units,    tenths    under 

tenths,  &c.,  and  proceed  as  in  Addition  of  Decimals.      (Art. 
187.) 

2.  What    is  the  sum  of   2358.35  liters,   861.45   liters, 
98.831  liters,  and  643.5  liters?  Ans.  3962.131  liters. 

QUEST.  —  22.  How  add  metric  numbers. 


14  METRIC    SYSTEM. 

3.  Find  the  sum  of  145.19  kilograms  of  sugar,  168.45 
kilograms,  431  kilograms,  8.60  kilograms,  36.1  kilograms, 
and  465.81  kilograms. 

4.  Find  the  sum  of  2346.43  meters  of  cloth,  45.3  rne- 
ters,  156.21  meters,  and  236.8  meters. 

Ans.  2784.74  meters. 

5.  What  is  the  sum  of  67.2560789  kilometers,  346.852 
decameters,  905.204  meters,  and  5670  millimeters. 

Ans.  71630.3699  meters. 

23.  Ex.  6.  From  6725.724  meters,  subtract  4.16631 
kilometers. 

Analysis. — Reducing  the  numbers  Operation. 

to  the  same  denomination,  4.16631  672"  724 

kilometers  =  4166.31  meters.  4166.31 

We  now  write  the  less  number  un-  12559  41 4        m. 
der   the  greater,  units  under  units, 

tenths  under  tenths,  <&c.,  and  proceed  as  in  ^Subtraction  of 
Decimals.     (Art.   189.) 

7.  What   is    difference    between    6843.94    liters     and 
394.203  liters  ?  Ana.  6449.737  liters. 

8.  Find   the   difference   between    931    kilograms    and 
391.275  kilograms. 

9.  Find  the   difference   between  6125   ares  and   61.54 
ares.  Ans.  606.346. 

10.  The    difference    between    563    myriameters     and 
265346  decimeters  ? 

11.  What  is  the  difference  between  568  steres  and  101 
decasteres  ? 

QUEST. — 23.  How  subtract  metric  numbers? 


METRIC    SYSTEM.  15 

Ex.  11.  How  much  silk  is  there  in  117^  pieces, 
each  of  which  contains  83.75  meters  ? 

Analysis. — Since  1  piece  contains 
83.75  meters,  Il7i  pieces  will  con-  Operation. 

tain  1174  times  as  much.  83.75     meters. 

We   multiply,  Und  point  off  the  117.5 

product  as  in  Multiplication  of  Deci-       9840.625  " 

mals.     (Art.  191.) 

12.  What  cost  4125.63  kiloliters  of  wheat,  at  $12.50  a 
kiloliter?  Ans.  51570.375  Kl. 

13.  What   cost  361   hectoliters  of  wine,  at  5.4  francs 
per  liter?  Ans.  194940  f. 

14.  How  many  square  feet  in  a  room  whose  length  is 
6.2  meters,  and  its  width  4.56  meters  1 

15.  At  $1.75  a  square  meter,  what  will  it  cost  to  carpet 
a  hall,  whose  length  is  16.5  meters,  and  whose  breadth  is 
7.4  meters?  Ans.  213.675. 

16.  If  1  are  of  land  cost  86.95  francs,  what  will  350.28 
ares  cost  ? 

17.  If  1   stere  of  wood  cost  6.25  francs,  what  will  79 
stercs  cost  1  Ans.  493.75  f. 

18.  What  will  it  cost  to  dig  a  cellar  12.2  meters  long, 
5.4  meters  wide,  and  2.8  meters  deep,  at   45  cents  per 
cubic  meter  ? 

19.  What  cost  65  hectares  of  land,  at  $15J  per  are  ? 

20.  How  many  ares  in  a  field  365  hectometers  long; 
and  243  decameters  wide  ? 

21.  What  will  7  hectoliters  of  brandy  come  to,  at  7.03 
francs  per  liter  ? 

QUEST.  —  24.  Hoar  mnltiply  metric  mm.bt-rs. 


16  METRIC    SYSTEM. 


Ex.  22.  A  man  divided  980.5  kilograms  of  flour 
equally  among  185  soldiers;  how  much  did  each  re- 
ceive! 

Analysis.  —  If  185    soldiers  re- 
ceive 980.5    kilograms,  1  soldier  Operation. 

must  receive  as  many  kilograms  as         185  )  980.5 
185  is  contained  times  in  980.5.  53      j^ 

We    divide,  and  point  off  the 
quotient  as  in  Division  of  Decimals.     (Art.  194.) 

23.  A   merchant  paid  $1872.40  for  234.45  meters  of 
broadcloth;  what  was  that  per  meter  ?       Am.  $7.986  -f- 

24.  Paid  216.15  francs  for  35.5  liters  of  molasses;  how 
inuch  was  that  per  liter  ? 

25.  A  man  traveled  5682.5  kilometers  in  74  days,  how 
far  did  he  travel  per  day  1  Ans.  757.66  +  Km. 

26.  How  many  cloaks  can  be  made  from  425.8  meters 
of  cloth,  allowing  to  each  cloak  15.4  meters. 

27.  A  farmer,  having  58.  65  ares  of  land,  wishes  to  fence 
it  into  fields  of  3.45  ares  each;  how  many  fields  can   he 
make?  Ans.  17  fields. 

28.  How  many  boxes,  each   holding  12.05  kilograms, 
will  it  take  to  pack  759.15  kilograms  of  butter  ? 

39.  The  area  of  a  grass  plat  is  21.06  sq.  meters,  and  its 
length  6.48  meters  ;   what  is  its  width  ? 

30.  If  f  pay  $276  for  92  meters  of  silk,  how  much  is 
that  per  meter  ?  Ans.  $3. 

31.  A  man  laid  out  3175  francs  in  flour,  at  25  francs 
per  barrel  ;  how  many  barrels  did  he  buy. 


QUKST.  —  25.  How  divi^e^Q^ric  numbers  ? 


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• 


f ractical^ntif  floipssi^  Cert-'^ooks. 


THOMSON'S  SERIES  OF  ARITHMETICS. 

Thi-  eminently  practical   and  scientific,  giving   the 

lores''  of  every  rule;   and  containing  much 
ot  found  in  other  works  of  the  kind. 

•,-•••:  (  :uplete,  and 

Vrithmetic-- 

TABLE   BOOK.     -For  Primary  School,-  ,}.). 

MENTAL  ARITHMETIC.    For  Begin  .a  En- 


RUDIMENTS  OF  ARITHMETIC; 

. 

ARITHMETICAL  ANA^Y2!3;    '  ik 

'' 

PRACTICAL  ARITHMETIC.  ...  •  •  j 

KEY  TO  PRACTICAL  ARITHMETIC 

largt-d.) 

HIGHER  ARITHMETIC?  or,  TJIK  s« 

OF    NlTMIii  

KEY  TO  HIGHER  ARITHMETIC 


The   most  !;leral  terms  for  first  suppli^  for   introduction,     • 
a,id  specimen  copies,  fvr  exatnination. 


